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Discrete series representations and K multiplicities for U(p, q). User's

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1. Remark that in some of the examples above it can happen that although the polynomials P and P displayed in the last table giving the A are different the polynomial P may coincide with P on J recall that several polynomials can have the same values on J N N Thus in this case we join the two intervals I and I and give only the polynomial P This streamlining of the function m i ts is given in the third column of the table describing the asymptotic behavior of m ttt We now give an example on U 3 4 Example 47 45 discrete 473 39 1 3 51 5 572 direction 1 0 0 0 0 0 1 gt function_discrete_mul_direction_lowest discrete direction 3 4 1451 20 t 1 120 t75 1 360 t76 851 360 t72 23 24 t73 5 36 t74 0 0 1431 12 t 19 8 t72 11 12 t73 1 8 74 1 37 3262622 2687514 5 t 73 240 t75 1 720 t 6 13275857 360 t72 64795 48 t 3 3977 144 t 4 38 39 265030 27790 t 1090 t72 20 t73 39 44 9631849 79305707 60 t 29 80 t75 1 720 t 6 3399664 45 t 2 110609 48 t 3 5675 144 t74 45 45 27182687 212385511 60 t 31 40 t 5 1 360 t 6 69073219 360 t 2 132929 24 t7 3 1619 18 t74 46 46 784945 886169 12 t 20959 8 t72 511 12 t73 1 8 t74 47 83 469370132 337238937 10 t 167 240 t75 1 720 t 6 363465857 360 t 2 774005 48 t 3 20897 144 74 84 85 5790400 235000 t 2820 t72 86 inf it ee Thus the multiplicity miis piecewise p
2. 1 2 3 4 1 3 2 3 in P v A A B 4 3 Valid permutations Let w We Remark that if wu A does not belong to the cone of non compact positive roots then the term corresponding to w in Blattner formula is equal to 0 It is important to minimize the number of terms in Blattner formula To this purpose we use a weaker condition we say that w W is a valid element if wp is in the cone spanned by all positive roots Thus if w is not valid the corresponding term to w in Blattner formula is equal to 0 As there is a simple description of the faces of cone spanned by all positive roots it is the simplicial cone dual to the simplicial cone generated by fundamental weights there is a simple algorithm that constructs valid permutations one at the time depending on the conditions they have to satisfy instead of listing all the elements of W The corresponding algorithm is used in 4 6 and we just reproduced it 5 Examples Example 45 We consider the discrete series representation indexed by and we test for the multiplicity mi where u is a K type We write HMowest for the lowest K type We use the algorithm whose command is gt discretemult p q 43 l m numeric case Croup mput A 31 2 15 2 9 2 5 2 3 2 5 2 owest 17 9 6 1 0 4 1017 1009 1006 1 999 1000 1004 p 100017 10009 10006 9999 10000 100004 31 2 19 2 11 2 15 2 7 2 37 2 Mlowest
3. 15 where A is the system of positive noncompact roots defined by a To simplify our next statements assume that G is semi simple and has no compact factors Then t is generated by non compact roots Recall that the spline function is given by a locally polynomial function on t well defined outside a finite number of hyperplanes see the description later Choosing the Lebesgue measure dh associated to the root lattice we may identify the measure Yat toa function denoted by Y Similarly we identify the measure dF to a function F gt on t Then we have almost everywhere the semi classical analogue of Blattner formula PA Y e w 4 wp A wEewe a and regular The anti invariant function F j restricted to the positive compact chamber ac is a non negative measure with support the Kirwan polyhedron Kirwan It follows from the study of spline functions that there exists a finite number of open polyhedral cones Rt in a x a so that the union of the cones R cover ax ac and polynomial functions p on a x a such that F u is given for a u ae u R by the polynomial pi A u on R A u de A u Rt In particular the Kirwan polyhedron Kirwan A is the union of the regions R A for which the polynomial p restricted to R A is not equal to 0 In fact the functions p A u are linear combinations of polynomial functions of w u where w are some elements of We If R i
4. Let us now consider the general case where F is no longer reduced to 0 For example for parabolic root systems of B Cr Dp the set F satisfying the condition C cannot longer be taken as equal to 0 We recall that an exponential polynomial function is a linear combination of exponential functions multiplied by polynomials Proposition 29 The function No h is an exponential polynomial function on V and the restriction of NY h to Vz is a quasipolynomial function on Vz 26 Proof Let us denote by p9 u 7 25 Y u the series development of the holomorphic function w9 appearing in kan 7 Then we see that JK K G h equals h u 9 e h27vVZIG JIR e ji a Hier S Z g r I g r k e h 2n 1G JK aa wi u DET AE r k Ces ai u plu The function lh u l r k 4 hw JK b gt Iron ai u Vk u is a polynomial function of h of degree I g r k Thus we see that JK K G h is the product of the exponential function e 2 V by a polynomial function of h Furthermore if g is of order p and h varies in Vz the function h gt e h 2nv 1G is constant on each coset h pVz of the lattice pVz Return to the computation of the partition function N4 h Thus we see that when h varies in c Z N Vz we have that N4 h coincide with the quasi polynomial function N h above Note that its highest degree component is polynomial and is again the function Y A the volume
5. compute preg DefVecne v At compute Ally P wpreg Ay compute conty ae an Ires K wu A pn end of loop running across w s collect all the terms and return m Y wcValid u w contw Figure 8 Blattner s algorithm numeric case Subroutines e Procedure to find At admissible hyperplanes e Procedure to deform a vector DefVecne v At e Procedure to compute M M CP v AT as in Fig 7 h e Procedure to compute Kostant function K h K 0 h Tn dae or more generally acat K g h Input Ao and vs for each H noncompact wall if H v 0 then skip H else if H Ao wo H U lt 0 then skip H else collect tH H Ao wo H v end of loop running across H s order list t s as to ti ts where to 0 ts 00 choose an interior point t in each interval ti 1 Compute polynomial on each ti ti 1 following the scheme Fig 8 and the ordered basis determined by t output the sequence of values m is valid on t ti 1 t E N Figure 9 Blattner s algorithm asymptotic case 48 References 1 10 Peat 11 Atlas of Lie Groups and Representations http atlas math umd edu Bliem T On weight multiplicities of complex simple Lie algebras Dis sertation Universitt zu Kln 2008 Baldoni M W Cochet C Beck M and Vergne M A polyno mial time algorithm for counting integral points in polyhedra when the dimension is fix
6. 51 2 5 2 155 2 direction 1 0 0 0 0 1 gt function_discrete_mul_direction_lowest discrete direction 3 3 1 24 t 74 5 12 t 73 35 24 t72 25 12 t 1 0 16 3059 2242 3 t 133 2 t 2 19 6 t 3 17 21 11914 9597 4 t 4367 24 t 2 27 4 t 3 1 24 t 4 22 40 100016 8664 t 228 t 2 41 inf That is for A discrete and u 30 20 1 26 2 79 the lowest K type 1 24 tt 5 12 t 35 24 t 25 12 xt 1 0 lt t lt 16 _ J 8059 2242 3 t 133 2 x t 19 6 t3 17 lt t lt 21 Mutt 11914 9597 4 t 4367 24 t2 27 4 t 1 24 xt 22 lt t lt 40 100016 8664 t 228 t t gt 41 The time to compute the example is TT 0 835 and the formula says for instance _ that m s9000008 911982672100016 To compare with other parametrizations of the discrete series representations it may be useful also to give here the command for the lowest K type of the discrete series with parameter discrete of the group U p q The command is gt Inf_lowestKtype p q discrete Example 8 gt Inf_lowestKtype 5 2 3 2 3 2 5 21 2 2 7 2 3 2 3 2 7 2 Similarly we may want to parametrize a representation of K by its highest weight Then the command is gt voganlowestKtype discrete p q Example 9 gt voganlowestKtype 5 2 3 2 3 2 5 2 2 2 3 1 1 3 Let us finally rec
7. Discrete series representations and K multiplicities for U p q User s guide Velleda Baldoni and Michele Vergne Abstract This document is a companion for the Maple program Discrete series and K types for U p q available on http www math jussieu fr vergne We explain an algorithm to compute the multiplicities of an irre ducible representation of U p x U q in a discrete series of U p q It is based on Blattner s formula We recall the general mathematical background to compute Kostant partition functions via multidimen sional residues and we outline our algorithm We also point out some properties of the piecewise polynomial functions describing multiplic ities based on Paradan s results Contents 1 The algorithm for Blattner s formula main commands and simple examples 4 2 Mathematical background 12 241 Notations es ea e Be ee ee se GS ee ee ee ak 12 2 2 Blattner formula and multiplicities 0 12 2 3 Polynomial behavior of the Duistermaat Heckman measure 15 2 4 Quasi polynomiality results ooa 17 2 5 Aim of the algorithm what can we do aoaaa aa 18 25L N merik Aor ee Re aR ee a ee ee 18 2 5 2 Regions of polynomiality 18 2 5 3 Asymptotic directions 0000000048 19 3 Partition functions the general scheme 3 1 3 2 3 3 3 4 3 9 3 6 3 7 Definitions fyi Se Ai et n cake Ge Boe Sa Gis ea eee ee Spline functions and Kostant partition functi
8. O lt i lt s where to 0 and t 41 oo Consider J N Z an interval in Z described by two integers a b with a ceil t and b floor ti 1 The interval J Z can also be reduced to a point Then we find exponential polynomial function P t on R such that m is equal to P t for t LAZ If particular y is an asymptotic direction if and only if the last quasipolyno mial P t does not vanish The algorithm is as follows For each consecutive value t 41 choose ur u trv with ti lt tr lt ti 1 Then move very slightly ur in u Then for each w E W wus o lies in a tope T for A o Then w The right hand side of this formula is an exponential polynomial function of tEZ The algorithm implementing this procedure is described in Fig 9 Let us remark that our algorithm implementation is for type A and thus the P are polynomials 39 4 Blattner s formula for U p q 4 1 Non compact positive roots With the notation of Section 1 we let G U p q and K Up x Uy be a maximal compact subgroup Let E be a p q dimensional vector space with basis e i 1 p q and V as in Ex 16 Let r p q 1 Consider the set of roots A fe ej 1 lt i lt j lt p q We then choose T to be the diagonal subgroup of U p q and identify t with E In this identification the lattice of weights is identified with Z T the element n1 Np4q giving rise to the character
9. collect all these MPNS s for vj and J end of loop running across I s collect all MPNS s for the hyperplane H by taking the product of P J v end of loop running across H s return the set of all MPNS s for all hyperplanes Figure 6 Algorithm for MPNS s computation general case In our program we run this algorithm for an element v not in any admissible hyperplane without knowing in advance if v belongs to the cone C At The algorithm returns a non empty set if and only if v belongs to C A 3 5 The Kostant function another formula for subsys tems of At In this article we will be using partition functions for lists At A B described in the Example 16 These lists are sublists of a system of type A7 with r p q 1 In residue calculation we can use change of variables and thus use a formula for which iterated residues will be easier to compute Let us describe this formula We will describe it for sublists eventually with multiplicities of a system A We take the notations of Example 16 Let be a sequence of vectors generating V and of the form e e 1 lt i lt j lt r 1 eventually with multiplicities Let m i lt j be the multiplicity of 33 the vector e ej in and define tj mjj41 mjr41 1 We recall our identification of V with R and of U V with R defined by duality In this way as we already observed the root e ej 1 lt i lt j lt r produ
10. t exp i01 exp iOp4q e19 eet aFeta The system of compact roots A is Ac e ej 1 lt i lt j lt p U e ej p 1 lt i lt j lt p q The system of non compact roots is An e ej 1 lt i lt p p 1 lt j lt p q Let A be the Harish Chandra parameter of a discrete series for G and u the Harish Chandra parameter of a finite dimensional irreducible representation of K Because discrete series are equivalent under the action of the Weyl group of K then we may assume that A a 8 where a 7 _ aiei a1 Qp 1 gt a2 gt gt Qp and B D Fi bal 61 gt b20 gt ba Here a 8 are integers if p q is odd or half integers if p q is even that is we fix as system of positive compact roots the system At e e 1 lt i lt j lt py Uf e e ptl lt i lt j lt ptq We parametrize u Pf by another couple H fa b a1 2 apl ieee bq with a gt gt ap and bi gt gt bg Here a are integers if p is odd half integers if p is even Similarly b are integers if q is odd half integers if q is even As the center of G acts by a scalar in an irreducible representation we need that the sum of the coefficients of A has to be equal to the sum of the coefficients 40 of u for the multiplicity of u in m to be non zero Thus p is in V see Ex 16 We now parametrize the different dominant chambers of t modulo the Weyl group of K by a subset A of
11. A p and B q Letr p q 1 We define At A B as the sublist of AF defined by A A B ei ej 1 lt i lt j lt r 1 with i Aj Ee Borie BEA These systems are the system of positive roots for the maximal parabolics of gl r 1 for different choices of orders Let us go back to the general scheme For any subset S of V we denote by C S the convex cone generated by non negative linear combinations of elements of S We assume that the convex cone C A is acute in V with non empty interior If S is a subset of V we denote by lt S gt the vector space spanned by S Definition 17 A hyperplane H in V is At admissible if it is spanned by a set of vectors of At When At A A or At A B then an At admissible hyperplane will be also called a noncompact wall Chambers Let Vsingl AT be the union of the boundaries of the cones C S where S ranges over all the subsets of At The complement of Ving A in V is by definition the open set Creg At of regular elements A connected component of Creg AT is called a chamber of C A Remark that in our definition the complement of the cone C A in V is a chamber that we call the exterior chamber The chambers contained in C A that we will call interior chambers are open convex cones Sometimes chambers are called cells or big cells by other authors The faces of the interior chambers span admissible hyperplanes The following pictures illustrate the situation
12. a i if eNC o 9 where o Bases At and vol c is the volume of the parallelotope 5 gt _ 0 lJa computed for the measure dh There exists a linear form JK that we call the Jeffrey Kirwan residue on R 4 such that JK takes the above values on the elements fs and is equal to 0 ona degenerate fraction or on a rational function of pure degree different from r If c is the exterior chamber then clearly JK is equal to 0 as is not contained in C AT We may go further and extend the definition of the Jeffrey Kirwan residue to the space R4 which is the space consisting of functions P Q where Q is a product of powers of the linear forms a and P X o Py is a formal power series Then we just define if Q is of degree q JK P Q m JK Pi r Q as the JK residue of the component of degree r of P Q 3 2 Spline functions and Kostant partition function Let us recall the formulae for the spline function Y4 and for N4 h Definition 19 Let c be an chamber contained in the cone C A Define the function Y on V by h e y JK 4 h Io More explicitly as JK vanishes outside the degree r we have N r Ys h wo Ik i 1 i We thus see Y h is an homogeneous polynomial on V The proof of the following theorem is immediate 10 23 Theorem 20 Let Y h be the multispline function associated to At Let c be a chamber contained in the cone C A We have for h c Ya h Y h
13. equal to k at z 0 then we can write f z Q z z where Q z is a holomorphic function near z 0 If the Taylor series of Q is given by Q z 722 9 qsz then as usual the residue at z 0 of the function f z EZX g qs2 is the coefficient of 1 z that is q 1 We will denote it by res o f z To compute this residue we can either expand Q into a power series and search for the coefficient of z or employ the formula 10 res nof 2 gg 40 z 0 30 We now introduce the notion of iterated residue on the space R 4 Let 7 aj Q2 Q be an ordered basis of V consisting of elements of AT here we have implicitly renumbered the elements of At in order that the elements of our basis are listed first We choose a system of coordinates on U such that a u u i A function R 4 is thus written as a rational fraction P u1 u2 plur U2 Ur P uat9 Ur Guim u where the denominator Q is a product of linear forms Definition 35 If Ry the iterated residue Ires of for T is the scalar Iresz resy 0F Su _1 0 TCSu 00 U1 U2 Ur where each residue is taken assuming that the variables with higher indices are considered constants Keep in mind that at each step the residue operation augments the homoge neous degree of a rational function by 1 as for example res o 1 xry 1 y so that the iterated residue vanishes on homogeneous elements RA if the ho
14. 17 11 6 7 2 20 1017 1011 1006 993 998 1020 100017 10011 10006 9993 9998 100020 11 2 7 2 3 2 1 2 9 2 5 2 1 2 3 2 U 3 3 2 5 2 1 2 7 2 11 2 5 2 1 2 2007 2 3 2 1 2 1 2 3 2 2 9 2 1 15 2 9 2 3 2 2 9 2 3 2 3 2 2 972 7 2 5 2 2 1 f 2 2 72 2013 2 211 2 29 2 21 2 203 2 2005 2 20007 2 1 1 3 a 2 0 2 HMiowest 7 4 1 2 5 11 2 5 2 1 2 1 2 1007 4 1 2 5 11 2 5 2 1 2 2007 2 7 2 3 2 1 2 5 2 9 2 5 2 1 2 3 2 7 2 8 5 2 1 4 6 3 0 3 6 Hlowest Time ee ee oY 97 sc oT sec J ee TOTS sec i osas 11700255 0 538 sec Ree eel 0365 sec 120495492015 3 493 sec Ss eee E OE ecoe 273 719 sec ee 3052 sec 120495492015 13 752 sec pean 225 el Sose k a ra aal u N 106 4 2 0 102 1104 2 0 2 104 T458704350546472381 163 104 sec Example 46 We consider the discrete oe representation indexed by A a direction y and we test for the multiplicity m gt algorithm whose command is gt function_discrete_mu_direction_lowest_ v p q For completeness we list Howest relative to each example 44 ptt where u Ulowest is the lowest K type We use the Morir tE N asymptotic case 1 2 13 21 2 17 2 2 3 14 1 0 0 1 1 5 0 0 1 59 39 51 7 156 U 2 3 Piowest l 121 2 7
15. L and L In this case A A B N Hz is the product of two systems AV ANL BOL x A ANL BOL and thus reducible e if L is of cardinal 1 then Hz is a noncompact wall In this case A A B N Hz is At AN L BOL and thus irreducible At this point to compute the MNPS or better as we explained the M s we can proceed as in Fig 6 The algorithm is outlined in Fig 7 We conclude with the following observation A necessary and sufficient con dition for the set MPNS v A A B to be non empty is that v belongs to the 42 cone generated by A A B As far as we know the equations of this cone are not known except in a few cases It is clearly necessary that v belongs to the simplicial cone generated by all positive roots To speed up the calculations we check this condition at each step of the algorithm We conclude with a simple example with p 2 q 2 and A 1 2 B 3 4 We follow the outline described in Fig 6 The highest non compact root is 0 e e4 There are 3 noncompact walls not containing the highest root L 1 4 1 3 We choose a vector v 4 3 2 5 not on any noncompact walls Then 1 4 1 3 are all such that v and 6 are on the same side For L 1 the v projection do not belong to the cone generated by A A B N Ay eg 3 3 e4 For L 4 we obtain the element M 1 2 3 4 1 3 2 3 in P v A A B For L 1 3 we obtain the element M
16. More precisely for A u E R Ao po N Pg x Py we have 12 mn J e w NR wA u pn wEWe The right hand side of Equation 12 is a quasi polynomial function of 4 u antisymmetric in u It takes positive values if u is dominant for At Recall that in this case the multiplicity of u on A is the absolute value of the function m above Sec 2 2 38 Of course the symbolic calculation above with the present approach is limited to very small examples Also we are not able to determine the largest domains where the function m is given by a quasi polynomial formula A H 3 7 3 Asymptotic directions We address now a simpler problem We have the same setting that in the previous section but we are now testing only the noncompact walls crossing in one fixed direction v Let uo o be given with Ao a N Py an Harish Chandra parameter and Ho ace N Py Let v ac be integral We will do the calculation of mr when Le Ho tv with t gt 0 is in the half line in the direction v In the application uo will be the lowest K type o pn of our discrete series 7 We compute the values ti where uo tU wo cross a non compact wall other than the ones which may contain the line uo tv wAg These are the values where the line u may cross the domains of quasipolynomiality described above We order this finite set of values 0 lt t lt tg lt ti lt lt ts Consider the interval J t ti41
17. Remark 21 According to Theorem 18 this theorem gives the formula for the volume V4 h of the partition polytope II 4 h Let us now give the residue formula for the number of integral points N 4 of the partition polytope IL4 h Consider the torus T U Uz where U is the dual vector space to V and Uz C U is the dual lattice to Vz If G U we denote by g its image in T For Bases A we denote by T c the subset of T defined by T o 9 E T e 27v IG _ 1 for alla o Ga representative of g U Uz The set T is a finite subset of T For G U and h V consider the Kostant function K G h on U defined by elh 2n 1G u K gt 6 G h u TE G env Gt Remark 22 If h Vz the function K G h depends only of the class g of G in U Uz The function K G h u is an element of R4 Indeed if we write I g fi 1 lt i lt Nie i2 v IG i then ht ag 7 K G h hny Ta ew 7 G h u e fp where w9 u is the holomorphic function of u in a neighborhood of zero defined by ai u 1 ee a ee kE aiu e ai 2ny 1G u sett A a Ae By taking the Taylor series of et ysI u at u 0 we see that the function u K G h u on U defines an element of Ry If is a chamber of C AT the Jeffrey Kirwan residue JK K g h is thus well defined 24 Definition 23 Let c be a chamber Let F be a finite subset of U We define the function Nu on V by NEE h vo
18. Sec 3 7 3 The Atlas of Lie Groups and Representations 1 within the problem of clas sifying all of the irreducible unitary representations of a given reductive Lie group addresses in particular the problem of computing K types of discrete series The multiplicities results needed for the general unitary problem is of different nature as we are going to explain Given as input and some height h Atlas computes the list with multiplic ities of all the representations occurring in T of height smaller than h But the efficiency in this setting is limited by the height In contrast the efficiency of our program is unsensitive to the height of A u but the output is one number the multiplicity of u in 7 It takes almost the same time to compute the multiplicity of the lowest K type of t fortunately the answer is 1 than the multiplicity of a representation of very large height Our calculation are also very sensitive to the rank p q 1 For other applications weight multiplicities tensor products multiplicities based on computations of Kostant partition functions in the context of finite di mensional representations see 4 6 2 1 The algorithm for Blattner s formula main commands and simple examples Let p q be integers We consider the group G U p q The maximal compact subgroup is K U p xU q More details in parametrization are given in Section 4 A discrete series representation T is parametrized accord
19. and Pj have to coincide on the intersection Secondly the intervals I NN can be reduced just to a point so that the polynomial F if not constant is not uniquely determined by its value on one point More generally if the length of the interval J is smaller that the degree of P the polynomial P is not uniquely determined Because of our focus on the polynomiality aspects we keep in the output of the algorithm the polynomials P even if the intervals I are reduced to a point or with small numbers of integral points In our application u will be the lowest K type and we will give some examples of the situations occurring in particular we will examine the following cases e The first example outputs a covering of N given by a unique interval and a polynomial function on N that computes the multiplicity Thus in this case all the integer points on the half line u tv t gt 0 give rise to K types that appear in the restriction of the discrete series in particular y is an asymptotic direction see Sec 2 4 and 3 7 3 e In the other examples the covering of N has at least two intervals and illustrate different situations To compute in the sense we just explained the multiplicity of the K type u t in the discrete series with parameter discrete u being the lowest K type moving in the positive direction y labeled by direction the command is gt function_discrete_mul_direction_lowest discrete direction p q E
20. compute Example 2 We consider the discrete series indexed by lambda33 11 2 7 2 3 2 9 2 5 2 1 2 Its lowest K type is lowlambda33 7 4 1 5 2 1 Of course the multiplicity of the lowest K type is 1 Our programm fortunately returns the value 1 in 0 03 seconds Consider now the representation of K with parameter biglambda33 10006 4 9998 10004 2 10000 Then the multiplicity of this K type is computed computed in 0 05 seconds as gt discretemult biglambda33 lambda33 3 3 2500999925005000 Here are two other examples that verify the known behavior of holomorphic dis crete series The notation ab that we use to label the discrete series parameters is introduced in 4 1 and it very effective to picture the situation but it is not relevant to understand the following computation Example 3 Consider a holomorphic discrete series of type aaabbb for G U 3 3 see Sec 4 1 with lowest K type of dimension 1 We verify that the multiplicity is 1 for u in the cone spanned by strongly orthogonal non compact positive roots hol33 11 2 9 2 7 2 5 2 3 2 1 2 lowho133 7 6 5 1 0 1 bigho133 7 1000 6 100 5 10 1 10 100 1 1000 gt discretemult lowhol133 ho133 3 3 1 TT 0 052 gt discretemult bigho133 h0133 3 3 1 TT 1 003 Example 4 Consider a holomorphic discrete series of type aaabbb for G U 3 3 see Sec 4 1 with lowest K t
21. group Lie algebra of G maximal compact subgroup of G Lie algebra of K maximal torus of K Lie algebra of T lattice of weights of T system of positive roots for G system of positive compact roots system of positive non compact roots system of positive roots of parabolic type determined by A B positive chamber for G and K respectively half the sum of positive positive compact positive non compact roots set of G admissible K admissible parameters set of G admissible and regular K admissible and regular parameters r dimensional real vector space x U dual vector space of U hE V the pairing between U and V lattice of V dual lattice in U a sequence of vectors in Vz a AT a tope torus U Uz t ET finite subset of T polytope defined by AF number of integral points for H 4 A chamber Jeffrey Kirwan residue iterated residue 2 2 Blattner formula and multiplicities Let G be a reductive connected linear Lie group with Lie algebra g and denote by K a maximal compact subgroup of G with Lie algebra We assume that the ranks of G and K are equal Under this hypothesis the group G has discrete series representations Recall Harish Chandra s parametriza tion of discrete series representations We choose a compact Cartan subgroup T C K with Lie algebra t Let P C t be the lattice of weights of T They correspond to characters of T Here if A P the corresponding character of T 1
22. in the wave front set it is sufficient to be approached in the projective space by lines R un with un Pf Na such that the multiplicity M is non zero and the sequence un is going to the infinity in ae Thus we do not know if a rational line uo tv contained in the Kirwan polytope could totally avoid the support of the function m We do not think this is possible 3 Partition functions the general scheme 3 1 Definitions Let U be a r dimensional real vector space and V be its dual vector space We fix the choice of a Lebesgue measure dh on V Consider a list At of non zero generators for V given by At gt a1 2 Ps ay We recall several results concerning partition functions that appear in 3 in the general context However let us describe right away the system of vectors At A B C A that will appear in our programs and that describe parabolic subsystem of A as we said the same method could be applied to other parabolic root systems Example 16 e Let E be an r dimensional vector space with basis e i 1 7 1 Consider the sequence At e e 1 lt i lt j lt r lj 19 This is a system of positive roots of type Ap We let V to be the vector space r l r 1 v fa Y neen Sh o i 1 i l Let Vz be the lattice spanned by A We may identify V with R by hw hi h2 hr In this identification the lattice Vz is identified with Z e Let A B be two complementary subsets of 1 2 3 p q with
23. on each coset of some sublattice L of finite index in L The subsets Py Py are shifted lattices and we may say that a function k on Py is quasi polynomial on P x P if the shifted function k A p pe is quasipolynomial on the lattice P x P Theorem 13 Let R be a domain of polynomiality in a x ac for the Duistermaat Heckman measure Then there exists a quasi polynomial function P on Py x Pe such that mi P u for any A pn E RN Py x PE AGa p Eac In fact the functions P are linear combinations of quasi polynomial functions of wA u where w are some elements of We The K types occurring with non zero multiplicity in m are such that u is in the interior of the Kirwan polyhedron Kirwan A In particular the lowest K type Hlowest 18 in the interior of Kirwan A In particular all the K types occurring with non zero multiplicity in m are such that u is in the interior of the cone A Cone A We believe they are contained in the cone Mowest Cone A but we do not know if this assertion is true or not by Vogan s theorem they are contained in Mowest Cone A If v t we say that v is an asymptotic direction if the line pjowest tu is contained in Kirwan for all t gt 0 The set of asymptotic directions form a cone which determines the wave front set of m K For the holomorphic discrete series the descrition of the cone of asymptotic diections is known In fact if the lowest K type of 7 is
24. roots of with respect to t and An the system of noncompact roots We let At be the unique positive system for A with respect to which A is dominant We write p Pacat O 5 Pe Tich a and pn PDR a where At A N Ae and At A N An Therefore P p P R pe P and if Py then pn P Write a ae C t for the closed positive chambers corresponding to At and At We will also write AT Af A and At A for At At Ay if necessary to stress that these systems depend on A Then for u Py N ac Blattner s formula says 1 md Y e w Pa wy A pn wEewe where given y t we define P y to be the number of distinct ways in which y can be written as a sum of positive noncompact roots recall our identification for which A A C t The number P 7 is a well defined integer since the elements of Af span a cone which contains no straight lines As usual e w will 13 stand for the sign of w Remark that as u pe and A pn pe are weights of T the element wu A pn is a weight of T It is convenient to extend the definition of m to an antisymmetric function on P As we observed already an element u Pf can be conjugated to a unique regular element in the corresponding positive chamber a C t for compact roots via an element w We Thus we define m e w m p Of course with this generalization the multiplicity of the K type u is Im and we can complete our picture for U
25. this paper addresses the question of computing m The algorithm described in this paper checks whether a certain K type 7 appears in the K spectrum of a discrete series 7 by computing the multiplicity of such a K type The input u can also be a symbolic variable as we will explain shortly By Blattner s formula computing the K multiplicity is equivalent to compute the number of integral points that is a partition function for specific polytopes We thus use the formulae developed in 3 to write the algorithm One important aspect of these results is that the input datas A p can also be treated as parameters Thus in principle given Ao 4o we can output a convex cone containing Ag 4o and a polynomial function of the parameters A u whose value at A 1 is the multiplicity of the K type pin the representation 7 as long as we stay within the region described by the cone and A u satisfy some integrality conditions to be defined later We also plan to explicitly decompose our parameter space A u in such regions of polynomiality in a future study thus describing fully the piecewise polynomial function m at least for some low rank cases In practise here we only address a simpler question we will fix A u and a direction y and compute a piecewise polynomial function of t coinciding with m 41g ON integers t This way we can also check if a direction v is an asymptotic direction of the K spectrum in the sense explained in
26. 1 2 r 1 of cardinal p Let B its complementary subset in 1 2 r 1 To visualize A B we write a sequence of lenght p q of elements a b with a in the places of A b in the places of B for example if A 3 5 and B 1 2 4 then we write b b a b a or simply bbaba Now we use this visual aid and describe a permutation w4 of the index 1 2 p q by putting the index 1 p in order and in the places marked by a and the remaining indices p 1 p q in order and in the places marked by b precisely wa 1 2 3 4 5 8 4 1 5 2 The elements w4 where A varies describe a system of representatives of Up q Lp X Yq being the permutations on n letters that is also the chambers of t for A g t modulo W In the above the chamber is described by h h1 h2 h3 h4 hs h3 gt h4 gt hy gt hs gt ho Let standard be the chamber a1 gt a2 gt gt Ap gt By gt 2 gt By Then if wadstandard we have AT A At and A A is isomorphic to At A B by relabeling the roots via w3 The next example will clarify the situation The subset A can be read from we reorder completely the sequence A and define A as the indices where the first p elements of are relocated Example 43 Let G U 2 3 with compact roots A e1 e2 3 4 3 5 e4 es and noncompact roots Ar e1 3 1 4 1 5 2 3 2 4 2 e5 Let A a 8 wit
27. 1 Numeric We have as input At a sequence of vectors in our lattice Vz a vector h Vz and we want to compute N4 h We will compute it by Na h Nahh Pn pr We mean Let 7 be any tope such that h h pn belongs to the closure of 7 Using Theorem 34 then Na h Nar h pn N44 R pr To compute a tope T containing h we can move h pn in any generic direction Here is an outline of the steps needed to compute the number N 4 h by the formula Na h N3 h Input a vector h Vz and A a sequence of vectors in Vz Output the number N4 h e Step 1 Compute the Kostant function e Tear T ESS or more generally compute a set F and the functions K G h for G F K h K 0 h e Step 2 Find a small vector so that if h is in Vz the vector h pn does not belong to any admissible hyperplane Thus the vector hreg h pn e is in a unique tope 7 The procedure to obtain hreg is called Def Vectorne h e Step 3 Compute the set All P hreg AT as explained in Fig 6 e Step 4 Compute N h by computing the iterated residues of K G h associated to the various ordered basis M for M varying in the set All That is compute the number out gt elh27v 1G So Ires K G h GEF MEAL where M is the ordered basis attached to M Then N4 h out 35 3 6 2 Symbolic The previous calculation runs with symbolic parameters If hfix is an element in Vz we
28. 2 is e Let At C P be a positive system of roots and p Sn hee a Then the subset p P C t does not depend of the choice of the positive system At We denote it by Py We denote by Py C Fy the subset of g regular elements We can similarly define Py We denote by W the Weyl group of K For any A Py Harish Chandra defined a discrete series representation m Elements of Py are called Harish Chandra parameters for G Two representations m and x co incide precisely when their parameters A are related by an an element of We Thus the set of discrete series representations is parametrized by Py We In the same way we can parametrize the set K of classes of irreducible finite dimensional representations of K by their Harish Chandra parameter u Py We Once a positive system of compact roots is chosen an element u Py can be conjugated to a unique regular element in the corresponding positive chamber a C t for the compact roots We denote by T K or simply by uE K the corresponding representation A discrete series representation T is K finite A X i A T Ik MT pe T EK is a basic problem To determine M that is the multiplicity of the K type 7 in m in representation theory Blattner s formula 14 gives an answer to this problem We need to introduce a little more notations before stating it We let A be the root system for g with respect to t Ae the system of compact roots that is the
29. 2 1 Fig 3 in the following way 63 Figure 3 m as antisymmetric function on U 2 1 The representation Tiowest With Harish Chandra parameter Hlowest Pn is the lowest K type of the representation m and occurs with multiplicity 1 It is in general difficult to compute mi for general p We will use Blattner s formula to compute m Our algorithm is based on a general scheme for computing partition functions using multidimensional residues Note that the presence of signs in Blattner formula doesn t even allow to say if a K type appears without fully computing its multiplicity Recall that if At is a positive root system of a semi simple Lie algebra the formula for the partition function has been used to compute tensor product de composition or weight multiplicities 4 6 14 2 3 Polynomial behavior of the Duistermaat Heckman measure Recall Paradan s results 16 on the behavior of the function m and its support For this it is useful to first recall the semi classical analog of Blattner formula Let us fix a positive system of roots At and consider the corresponding closed positive chambers a ac C t for At and At Our parameter A varies in a and it is non singular In this subsection the integrality condition A Py is not required Let Oy C g be the coadjoint orbit of A It is a symplectic manifold and thus is provided with a Liouville measure dG Let p O amp be the projecti
30. 9 2 51 6 157 1 0 0 0 1 A 341 2 49 2 3 2 5 2 11 2 371 2 Hlowest 345 2 53 2 5 2 7 2 13 2 373 2 A 1343 2 31 2 21 2 13 2 19 2 363 2 Hlowest 173 17 12 8 11 183 0 ae e ee 52 020 sec t 173 if t gt 174 E A _ jf t 1 if O lt t lt 155 1 1 0 0 1 1 156 if t gt 156 7 1730 sec A1 1 2 t 1 2 t 1 0 O 1 1 2 t 3 2 t 1 1 0 2 in fT A2 15 2 t 7 2 t 1 0 OJ 1 1 101 66 1 2 t 23 2 t 11 11 0 12 in f J A3 1 2 t 1 2 t 1 0 0 1 1 wn f A4 t 1 0 in f A5 1 10 3 t 7 2 t 25 6 x t3 0 0 1 1 3 t 1 2 t 1 6 t 1 1 1 1 2 6 7 2 x t 1 2 A 2 3 0 3 inf T 13 4 t 51 8 t 61 4 t 89 8 t7 0 0J 1 3 2 t 1 2 t7 1 I 3 27 4 t 1 8 t 3 4 t3 1 8 t4 2 3 6 4 170 32664996 9380059 12 t 168227 24 t 335 12 t3 1 24 t4 171 172 15225 349 2 t 1 2 t 172 173 78155000 10718575 6 t 183749 12 t 175 3 t3 1 12 t4 174 175 0 176 in f T 5 4 t 1 24 t 1 4 t 1 24 t7 0 0 lt 1 1 154 23726781 2457885 4 x t 143207 24 t 103 4 t 1 24 t4 155 156 156 156 inf
31. a one dimensional rep resentation of K the exact support of the function m has been determined by Schmid We will explain in Section 3 7 how to compute regions of polynomiality R and the quasi polynomial P A 17 The quasi polynomials P are in some particular space of quasi polynomials satisfying some system of partial difference equations For a a non compact root consider the difference operator V acting on Z valued functions on P by Vak u k n k u a Definition 14 A quasi polynomial L on P is in the Dahmen Micchelli space DM A if p satisfies the system of equations I Va b 0 ac AT Q for any Q Hy Then the following result follows from Dahmen Micchelli theory of partition func tions Proposition 15 The quasi polynomial u P A u belongs to the space DM Ax 2 5 Aim of the algorithm what can we do Our algorithm addresses the following questions for U p q All of these questions will be analyzed in more details in Sec 3 7 2 5 1 Numeric We enter as input two parameters A Py x Py The output is the integer M see Sec 3 7 1 2 5 2 Regions of polynomiality The input is two parameters Ag uo Py x Py Let a ac be the chambers determined by Ag and uo We also give two symbolic parameters A ju Then the output is a closed cone R Ag Wo C a ae described by linear in equations in A u containing Apo 4o and a quasi polynomial P in A p such that mi P A n
32. all the simple case of the multiplicity function for G U 2 1 see Sec 4 1 for the notations Choose as positive compact root the root e e2 and denote by w the corre sponding simple reflection Fix A A1 2 A3 the Harish Chandra parameter for a discrete series representation We assume A regular and A gt A2 There are three chambers cj c2 3 and hence three systems of positive roots containing e1 e2 Precisely c corresponds to the positive system e1 2 1 3 2 e3 c2 corresponds to the positive system e1 e2 e 3 e3 e2 and c3 to e1 2 3 1 3 e2 We examine the situation in which the discrete series parameter belongs to one of these chambers Fig 1 and Fig 2 picture the two chambers 1 cz and evidentiate the values for mi when is in the chamber The black lines mark the chambers containing the compact root e1 e2 and the red ones the system of positive roots for the given chamber 2 amp 3 Figure 1 m for the chamber c of U 2 1 10 e1 3 e3 2 Figure 2 m for the chamber cz of U 2 1 We give some examples concerning the two situations The parameter A in Fig 1 is hol21 2 1 3 In Figure 2 the parameter A is aba 2 3 1 11 2 Mathematical background 2 1 Notations a de P PT Pr P Pi Pr Pr U V Col Vz Uz At JK Tres semi simple real Lie
33. asic formula for our calculations Theorem 39 12 Let c be a chamber and let v c Then for Rat we have 1 JK y aa a MEP v AT Let us finally sketch the algorithm to determine P v A without going to construct all the MNS s If At Af x Aj is reducible then P v At is the product of the corresponding sets P v AT 32 Assume A is irreducible Let 0 be the highest root of the system A for our order ht We start by constructing all possible A admissible hyperplanes H for which v and 6 are strictly on the same side of H In particular the hyperplane H does not contain the highest root Then we compute the projected vector projyv on H parallel to 0 v proj p v t with projy v H and t gt 0 and compute At N H If My is in P projyv ATN H then M op My A is in P v AT Running through all hyperplanes H for which v and are strictly on the same side of H we obtain the set P v AT Let us summarize the scheme of the algorithm in Figure 6 Recall that we have as input a regular vector v and as output the list of all MPN S s belonging to P v A for each hyperplane H do check if v and 0 are on the same side of H if not then skip this hyperplane define the projection projz v of v on H along 0 write AN H as the union of its irreducible components I U U Ik write v as v1 PB vz according to the previous decomposition for ach I do compute all MPNS s for vj and I
34. ces the linear function u uj on U while the root e e 4 produces the linear function u We are now ready to give another formula for the Kostant function in this situation Theorem 40 Let c be a chamber of C Let h Yit hie hi hr then N h vol V Vz dh JK fo h u where gt 1 ae ug rite i fah ______ Il u i r 1 II ui uj ii i 1 I lt i lt j lt r Thus when h Vz A c Z we have No h Ng h Example 41 e If AF then E Hea a 7 Ti lt icjcp us uj x i ui e Let p q integers such that p q r 1 and be the system of positive noncompact root for A defined by e e j 1 lt i lt p p 1 lt j lt r 1 That is A A B with A 1 p and B p 1 p ql Then falhi h2 hr u 1 uj ta 1 Up rr 11 1 ne Up4i ret1 t 1 Upt q 1 Pta 1 1 u1 Up i U1 Uptg 1 Up Ups Up Up q 1 U1 2 Up Proof The function K 0 h u e T eg 1 e7 computed for the system is efit eh2u2 ehrur K 0 h u 1 enti Misr 1 I 1 n e uius 6 9 i 1 1 lt i lt j lt r Note that the change of variable 1 z e preserves the hyperplanes u 0 and u uj and that z e 1 leads to dz e du 1 du Thus after the change of variable we get the required formulae 34 3 6 Computation of Kostant partition function gen eral scheme 3 6
35. ed Math Oper Res 19 1994 769 779 Cochet C Multiplicities and tensor product coefficients for A 2003 available at math ArXiv CO 0306308 Brion M and Vergne M Residue formulae vector partition func tions and lattice points in rational polytopes J Amer Math Soc 10 no 4 1997 797 833 Cochet C Vector partition functionsand representation theory 2005 available at math ArXiv RT 0506159 Dahmen W Micchelli C The number of solutions to linear Dio phantine equations and multivariate splines Trans Amer Math Soc 308 no 2 1988 504 532 Jeffrey L C and Kirwan F C Localization for nonabelian group ac tions Topology 34 1995 291 327 Szenes A and Vergne M Residue formulae for vector partitions and Euler MacLaurin sums Advances in Applied Mathematics 30 2003 295 342 Baldoni Silva W and Vergne M Residues formulae for volumes and Ehrhart polynomials of convex polytopes manuscript 81 pages 2001 available at math ArXiv CO 0103097 Baldoni Silva W De Loera J A and Vergne M Counting In teger flows in Networks Foundations of Computational Mathematics 4 2004 277 314 available at math ArXiv CO 0303228 49 12 De Concini C and Procesi C Nested sets and Jeffrey Kirwan cycles available at math ArXiv AG 0406290 v1 2004 13 Khovanskii G and Pukhlikov A V A Riemann Roch theorem for integrals and sums of quasipolynomials over virtual polytopes St Pe
36. for any A u R Ao Ho N P3 x Pe We worked out part of this program for U p q but it is still not fully imple mented In particular for the moment we are not able to produce a cover of a x ae by such regions The number of regions needed grows very fast with the rank Furthermore we are not able to decide when we have finished to cover a x ae 18 2 5 3 Asymptotic directions We implemented for U p q a simpler question which gives a test for asymptotic directions Let s consider as input parameters Ag in Py and a weight v ae Let uo be the lowest K type of 1 The line t Xo Ho tU cross domains of polynomiality R at a certain finite number of points 0 lt t lt tg lt lt ts Let us define to 0 ts41 co Then we study the function P t ip ae tEeN t gt 0 We can find polynomials qi 4 of degree bounded by pq p q 1 such that P t a 4 t when t lt t lt ti fori 0 s and t N In particular the direction v belongs to the asymptotic cone of the Kirwan polyhedron if and only if our last polynomial qj gt o is non zero see Sec 3 7 3 As we discussed in the first part if the intervals are two small these polyno mials are not uniquely determined However the last interval is infinite and the last polynomial is well determined If this last data is non zero then v is in the wave front set of t The reciproc is not entirely clear Indeed for a direction to be
37. for the interior chambers in the case of A Fig 4 and the interior chambers for various subsystems of Aj of type AtT A B relative to U 2 2 Fig 5 20 e2 amp 3 j e3 e2 4 e1 e2 e3 4 Figure 4 The 7 chambers for AF e2 amp 3 e1 6 amp 3 e2 amp 4 NEZ e1 amp 2 e3 4 e1 amp 4 A 11 4 2 3 A 1 2 3 4 ey 8 aa ey amp 2 e3 e4 A 1 3 2 41 Figure 5 Parabolic subsystems of AZ 21 Polytopes We consider the space RY with its standard basis w and Lebesgue measure dz If z DA riwi RN we simply write xz 1 y Consider the surjective map A RY gt V defined by A w aj If h V we define the convex polytope II4 h consisting of all non negative solutions of the system of r linear equations ai 1 via h that is I h 2 a1 0n ERY Ar h zi gt 0 We call IL 4 h a partition polytope associated to At and h We identify the spline distribution Y4 by Formula 3 to a function still denoted by Y4 using dh Recall the following theorem which follows right away from Fubini theorem 10 11 Theorem 18 The value of the spline function Y at h is the volume of the partition polytope IL4 h for the quotient measure dx dh The spline function Y4 is given by a polynomial formula on each interior chamber It is identically equal to 0 on the exterior chamber Parti
38. h a 4 2 4e 2e2 and 8 6 5 3 6e3 5e4 365 Then A 3 5 B 1 2 4 that is the configuration bbaba The system of non compact positive roots for A is e3 e1 4 e 1 e 1l e5 e3 2 4 2 5 2 isomorphic to At A B by the relabeling of the roots suggested by was that is e3 fi ea fo e1 f3 5 fa 2 fs Thus relabeling the roots our calculations will be done for A A B inside Ay where A A B is given in Example 16 Remark here that A A B is irreducible if p and q are strictly greater than 1 In contrast when p or q 1 the system is fully reducible Consider for example the case p 1 Example 44 In this case A has only 1 element and the system A A B has r elements and is a base of V Thus At A B is fully reducible in the direct sum of p q 1 one dimensional systems For example take U 1 r with A 1 and B 2 r 1 Then At A B e1 e2 1 3 1 r41 is isomorphic to AT x AT xXx Af 41 Remark that when A1 A2 have the same number of elements although the sys tem of noncompact roots A A1 B1 and A Ag B2 are clearly isomorphic the combinatorial properties of A A B may vary For example see Figure 5 if A 1 2 B 3 4 the cone generated by the non compact roots has basis a square and is not a simplicial cone If A 1 3 and B 2 4 then the cone generated by the non co
39. hn2nv 1G e Iresz K G wu pn GEF MeAllw ENDs 37 e Finally compute Ss e w x contributiony wEWe Let us comment briefly If w fireg A is not in the cone generated by non compact positive roots the set All is an empty set In particular we may restrict the computation by diverse consideration to valid permutations which have some chance to give a non empty set see Section 4 3 3 7 2 Symbolic The preceding calculation runs with symbolic parameter and we take advantage of this to find regions of polynomiality Let s explain how Let a be a chamber in t for the system A of roots of g Let U be the open set of A u a x t such that w u does not belong to any non compact wall Let Ao Ho E U Then we define R Ao po to be the closure of connected component of U containing Ao uo This region is a cone in t x t with non empty interior and can be described by linear inequalities in A ju For this domain we can compute a polynomial formula and state the following result If A varies in a the systems A A A A determined by remains the same We denote it by A At Furthermore if A p E R Xo Ho for any w We the element wd yu lies in a tope T for the system A which depends only of w Theorem 42 The domain R o To is a domain of polynomiality for the Duistermaat Heckman measure and thus is a domain of quasi polynomiality for the multiplicity function m
40. ing to Harish Chandra parameter A that we input as discrete discrete A1 lt Apl D15 gt Yall Here yj are integers if p q is odd or half integers if p q is even They are all distinct Furthermore A gt gt Ap and 7 gt gt Yq A unitary irreducible representation of K that is a couple of unitary irreducible representations of U p and of U q is parametrized by its Harish Chandra pa rameters u that we input as Krep Krep a1 a2 ap b1 55 89 with aj gt gt ap and bj gt gt by Here a are integers if p is odd half integers if p is even Similarly b are integers if q is odd half integers if q is even As we said our objective is to study the function mi for u K where u Krep and A discrete The examples are runned on a MacBook Pro Intel Core 2 Duo with a Processor Speed of 2 4 GHz The time of the examples is computed in seconds They are recorded by TT Some of these examples are very simple and can be checked by hand as we did to reassure ourselves Other examples are given at the end of this article To compute the multiplicity of the K type given by Krep in the discrete series with parameter given by discrete the command is gt discretemult Krep discrete p q Example 1 Krep 207 2 3 2 3 2 207 2 discrete 5 2 3 2 3 2 5 2 gt discretemult Krep discrete 2 2 101 Here is another example of mi that our program can
41. ion functions Let us consider as before our lattice Vz and our sequence AF of elements of Vz Let 1 mai Da acAt 29 We introduce Pr Pn Vz Thus for any u Ph the function N4 pn is well defined Let H be the complement of all admissible hyperplanes that is hyperplanes generated by elements of At Def 17 Definition 33 A tope is a connected component of the open subset V H of V We choose once for all a finite set F of elements G of U so that the image of elements g cover all groups T o If T is a tope then 7 is contained in a unique chamber c and we denote by N the exponential polynomial function NY given in Definition 23 If 7 is not contained in C A then N 0 The closures of the topes 7 form a cover of V A consequence of Theorem 24 is the following Theorem 34 For any tope T such that u E TN Pa we have Nat u Pn Na H Pn 3 4 A formula for the Jeffrey Kirwan residue Having stated a formula for partition functions or shifted partition functions in terms of JK we will explicit it using the notion of maximal proper nested sets as developed in 12 and the notion of iterated residues The algorithmic implementation of this formula is working in a quite impressive way at least for low dimension This general scheme will be then be applied to Blattners formula 3 4 1 Iterated residue If f is a meromorphic function of one variable z with a pole of order less than or
42. l V Vz dh X IK K G h GEF where vol V Vz dh is the volume of the fundamental domain of Vz for dh Finally introduce the zonotope Z A to be the convex polyhedra defined by N Z A DD ti amp i 0 lt ti lt 1 i 1 When A is fixed we just write Z Z A and if C is a set we denote by C Z the set of elements z where C and z Z The following theorem is due to Szenes Vergne 9 It generalizes 7 13 and 5 Theorem 24 Letc be a chamber Let F be a finite subset of U Assume that for any o Bases A such that c C C o we have T o C F Uz Then for h VZN c Z we have Na h NSE h We choose for any chamber such a finite set F such that all elements g F Uz have finite order and such that F satisfies the condition C for any o Bases A such that C C c we have T a c F Uz It is possible to achieve this for example choosing a set F of representatives of Uz modulo Uz where p is that pUz is contained in gt Za for any basis 1 0 We now simply denote N by N leaving implicit the choice of the finite set F Remark 25 e Observe that N h does not depend on the measure dh as it should be e Ifc is the exterior chamber then Ns h 0 In our algorithm we are not knowing in advance if the point h belongs to the cone C A or not so that this remark is not as stupid as it looks e Observe also that if c is an interi
43. matical background We recall Blattner s formula and we discuss the piecewise quasi polynomial behavior of multiplicities for a discrete series of a reductive real group In the third part we outline the algorithm of computing partition functions for arbitrary set of vectors based on Jeffrey Kirwan residue and maximally nested subsets In the fourth part we spe cialize the method to U p q We in particular explain how to compute maximally nested subsets of the set of non compact positive roots of U p q In the last two sections we give more examples and some details on the im plementation of the algorithms for our present application Here G is the Lie group G U p q and K U p x U q is a maximal compact subgroup of G Of course all these issues can be addressed for the other reductive real Lie groups following the same approach described here To introduce the function we want to study we briefly establish some notations see Sec 2 2 We denote by m a discrete series representation with Harish Chandra parameter The restriction of m to the maximal compact subgroup K decomposes in irre ducible finite dimensional K representations with finite multiplicities in formula Xr Xr T Ik 7 MutT Tu where we sum over K the classes T Of irreducible finite dimensional K represen tations u being the Harish Chandra parameter of T m is a finite number called the multiplicity of the K type 7 or simply of u in T and
44. might want to find a tope 7 such that hfix belongs to the closure of T Then Na h Na h will be valid whenever h is in the closure of 7 Here is the outline of the algorithm Input hfix is an element in Vz and At a sequence of vectors Output A domain D C V and an exponential polynomial function P h on V The domain D is a closed convex cone in V described by linear inequations such that h fix isin D The formula P h N4 h is valid whenever h DA Vz e Step 1 Consider h as a parameter and compute the Kostant function K h u as a function of h u given by e lt hu gt Ileca l e lt au gt or more generally compute a set F and the functions K G h u for G F as function of h u K h u K 0 h u e Step 2 Find a small vector e so that if hfix is in Vz then the vector hfi reg hfix pn does not belong to any admissible hyperplane Compute the domain D F where 7 is the unique tope T containing h fi reg e Step 3 Compute the set All P hfiz eg AT as explained in Fig 6 e Step 4 Compute N h by computing the iterated residues of K G h associated to the various ordered basis M for M varying in the set All here h is treated now as a parameter That is we compute out Ss eh2nv 1G gt Ires K G h GEF MEAL where M is the ordered basis attached to M The output out is an expo nential polynomial function P h of h and once again we compute Na h P h ou
45. mogeneous degree of is different from r Observe that the value of Iresz depends on the order of 7 For example for f 1 y x we have resz oresy o f 0 and resy presz o f 1 Remark 36 Choose any basis 1 Y2 Yr of V such that laj Blay for every1 lt j lt r and such that y NY2 A Ar a1 AQa2A Aa Then by induction it is easy to see that for E R4 TeSa 0 TESq 0P TeSy 0 TESy 09 Thus given an ordered basis we may modify ag by ag cai with the purpose of getting easier computations As for the usual residue the iterated residue can be expressed as an integral as explained in 3 This fact allows change of variables 3 4 2 Maximal proper nested sets adapted to a vector We recall briefly the notion of maximal proper nested set NPS in short and some of their properties see 12 A subset S of At is complete if S S M At here recall that 9 is the vector space spanned by S A complete subset S is called reducible if we can find a decomposition V Vi V such that S S U So with S C Vy and S2 C Va Otherwise S is said to be irreducible 31 A set M Ih Ig I of irreducible subsets of A is called nested if given any subfamily 1 Jm of M such that there exists no i j with I C Jj then the set J U UIm is complete and the elements J are the irreducible components of h Ulg U UTI Then every maximal nested set M MNS in sho
46. mpact roots is the simplicial cone generated by e1 e2 e2 3 3 4 4 2 Algorithm to compute MPNS the case of A A B With the notations of Ex 16 we denote by A a proper subset of 1 2 7 1 with r p q 1 and by B the complementary subset to A in 1 2 7 1 Given v V and not on any admissible hyperplane we describe the algorithm to compute P v At A B If p or q 1 roots a in At A B form a basis on V thus there is only one maximal nested set M a a At A B Thus P v A A B is empty or equal to M depending if v belongs to the cone generated by At A B or not This is very easy to check If p gt 1 and q gt 1 we determine the set P v AT A B by induction going to admissible hyperplanes If L 1 2 r 1 is a proper subset of 1 2 7 1 we will also use the notation L fi L 1 lt i lt r 1 for the complement of L We denote by Hr v V Jier vi 0 the hyperplane determined by L the hyperplane Hy is equal to the hyperplane Hr determined by K It is very simple to describe A A B admissible hyperplanes that is noncompact walls The description is an adaptation of the A admissible hyperplanes that appear in 3 Keeping A fixed with A 1 r we consider hyperplanes Hzr indexed by subsets L C 1 2 7 1 with the following properties e if L A 1 or r then Hz is a noncompact wall if and only if both A and B intersect
47. n the closures of the chambers 28 The second example treats the unimodular case of A7 see Example 16 Since we have identified V with R then we have a canonical identification of U V with R defined by duality u R to u oi uje E where et is the dual basis to e Thus the root e e 1 lt i lt j lt r produces the linear function uj uj on U while the root e e 4 1 produces the linear function u Recall also the identification h YH hye ha hrl We compute the number of integral points for the parabolic subsystems of U 2 2 illustrated in Fig 5 Example 32 We consider the 3 different systems of non compact roots as de scribed in Fig 5 and give the formulae for the partition function 1 If A At 1 4 2 3 then hy h2 1 if hec _ h h h3 1 if h ca Nat h h h3 1 if he K Rr ET if hEca 2 If A A f1 2 3 4 then 1 ho if hea 1 hi h2 h3 if hea NAS ihi oe eee 1 hg if hEc4 3 If A AT 1 3 2 4 then 1 hi h if hea Nat h ITt hit thet h3 if h ca is 1 hi if h os We have to compute the Jeffrey Kirwan residue of the function f fx a where fi h w Iacat Sta x evihitushatushs The aEAn computation is immediate since we need only term of degree one for the expansion of f We omit the details Remark though that once again the formulae agree on walls as it should be 3 3 Shifted partit
48. nd 9 are on the same side then Hy if not skip the hyperplane define the projection v proja v of v on Hy along 6 compute v1 A1 11 v2 A2 I2 where Ay LN A Ag L N A h L Ip L and v1 v2 are the components of v on L and L respectively if vi resp v2 is not in the positive cone for AM then skip the hyperplane if A 1 apply the induction and compute M1 M E MNPS v A1 I add to Mj the root 67 do the same if A2 1 else apply the induction and compute ay Mi E P u Ai Li i 1 2 do the cartesian product M x M2 and add to each set the root 0r collect all M s for the wall L end of loop running across L s end induction return the set of all M s M MPN for all hyperplanes Figure 7 P v AtT A B 6 2 Numeric The scheme is described in Fig 8 6 3 Asymptotic directions We fix the parameter Ap and uo regular in the chambers a a and a weight v We want to compute tt In the application uo will be the lowest K type The scheme is described in Fig 9 47 Subroutines e Procedure to find A admissible hyperplanes e Procedure to deform a vector DefVecnc v At gt e Procedure to compute M M P v At as in Fig 7 eh e Procedure to compute Kostant function K h K 0 h To Ge acaz or more generally K g h e Compute the valid permutation Valid u v C We Input A u Compute At At A Compute Valid A u for each w Valid A p
49. of the polytope II Las h The quasipolynomial nature of the integral point counting functions N 4 stems precisely from the root of unity in formula 9 Furthermore for parabolic root systems of type B C and D these roots of unity are of order 2 as in the following example Thus we summarize the properties of our partition functions in the following remark Remark 30 e A is unimodular that is we can choose F 0 in Theorem 24 and thus the partition function Na for any subset of AF coincide with a polynomial function on each domain c Z e The integral point counting functions Ne for any subsystem of B C Dr coincide with quasipolynomials with period 2 on each domain c Z We now compute the number of integral points in two different situations a non unimodular case and a unimodular one We treat the non unimodular case first Example 31 Here V is a vector space with real coordinates and basis e1 e2 and U V has dual basis e e We write v par vje E V and u De he U 2 for elements in V and U respectively Let us compute the number of integral points for the positive non compact root system occuring for the holomorphic discrete series of SO 5 C that is we fix At e1 eo 1 2 1 e2 and At A e1 e1 2 61 e2 We also write a vector h hye hoez in the cone C AT as hi h2 Of course the calculation can be done by hand but we illustrate the method in this ve
50. olynomial function can be completely described by the following 1 31 12 t 1 8 t 11 12 t 19 8 t if 0 lt t lt 39 _ J 265030 27790 t 20 t3 1090 t if 40 lt t lt 46 Mutt 784945 886169 12 t 1 8 t 511 12 t 20959 8 t if 47 lt t lt 85 5790400 235000 t 2820 t if 86 lt t The time to compute the example is TT 19 487 and the formula says for instance that for A discrete and u 475 40 0 103 2 9 2 5 2 1147 2 the lowest K type then M 200000007 1127995300005790400 6 The program Discrete series and K mul tiplicities for type A We give a brief sketch of the main steps for the algorithms involved in Blat tner s formula 6 1 MNPS non compact We outline the algorithm that computes directly M for M P v At A B We are taking advantage of the fact that we know the M s in the case of A 1 r as we saw in Ex 44 In the following scheme p q are integers A C 1 2 p q is a set of cardinality p B is the complement subset defining U A B Or is the highest noncompact root for J If L C 1 2 p 4 we denote by L the complement set 46 Input v A I v a vector and A C I 1 2 p q A p proceed by induction on the cardinality of A if A 1 or A r write the unique M M MNPS determined by the situation if v C M then the output is M construct the hyperplane Hy check if v a
51. on Shifted partition functions 2 0000048 A formula for the Jeffrey Kirwan residue 3 4 1 Iterated residue 000 eee eee 3 4 2 Maximal proper nested sets adapted to a vector The Kostant function another formula for subsystems of A Computation of Kostant partition function general scheme IG ONUMETICN co ak Goa diese dae a aw a Mag a Ao 3 6 2 Symbolice a ga feck teks BAe Sa bee ee Computation of Blattner formula general scheme utils NNEC ose eaten is ae he Be ee eR Ree ae te EA B2 SOyMboOle v soans Sie Sete Boy a hoe ed a ae we Ee bs 3 7 3 Asymptotic directions 2 000048 4 Blattner s formula for U p q 4 1 Non compact positive roots 2 0004 4 2 Algorithm to compute MPNS the case of AT A B 4 3 Valid permutations 0 0 0 000000 0004 5 Examples 19 19 23 29 30 30 31 33 35 35 36 37 37 38 39 40 40 42 43 43 6 The program Discrete series and K multiplicities for type A 46 6 1 6 2 6 3 MNPS non compact oaa a INUMEPIC e jodi 2 ave E i a a ea ee a a eS we Asymptotic directions 0 a a Introduction 46 The present article is a user s guide for the Maple program Discrete series and K types for U p q available at http www math jussieu fr vergne In the first part we explain what our program does with simple examples The second part sets the general mathe
52. on This is a proper map Each coadjoint K orbit in intersect ae C t C Thus the projection of O on is entirely determined by its intersection with ae We recall that the set p O N de is a closed convex polyhedron Definition 10 The Kirwan polyhedron Kirwan A is the polyhedron p O N ae As far as we know there are no algorithm to determine the Kirwan polyhedron A weak result on the support of Kirwan is that Kirwan A is contained in A Cone A where Cone A is the cone generated by positive non compact roots The push forward of the measure dG along the projection p O gives us an invariant positive measure on By quotienting this measure by the signed Liouville measures dpp of the coadjoint orbits Ky in we obtain a W anti invariant measure dF on t More precisely for a test function on 2 SPAB a f denh f dBA CJt H Ox Here e is the locally constant function on t anti invariant by W and equal to 1 on the interior of the positive chamber ae We refer to dF as the Duistermaat Heckman measure If At a ay is a sequence of elements in t spanning a pointed cone the multispline distribution Y4 is defined by the following formula For a test function on CO CO N 3 Y4 f f OCS tiai dt dtn Then for a and u E t we have the following result due to Duflo Heckman Vergne 15 4 aF u Y e w w dy Yas wEeWwe
53. or chamber then c Z contains the closure t of c while if c is the exterior chamber c Z c For an interior chamber usually the set c Z intersected with the lattice Vz is strictly larger than intersected with Vz This fact will be important for computing shifted partition functions as we will explain later 25 Let us explain the behavior of the partition function N 4 on the domain Z We first explain the case of an unimodular system Definition 26 The system A is unimodular if each o Bases A is a Z basis of Vz Example 27 It is easy to see that A7 is unimodular so is any subsystem Thus if A is unimodular the set F 0 satisfies the condition C and we choose this set F Proposition 28 If At is unimodular the function Noe h is a polynomial func tion on V Proof We have just to consider K G h K 0 h and we can write h u h u N kanosen ee ee La e aiu I iu Ika e7 aiu m e aas en aj u hood of 0 with yolu It follows that N 7 is given by the following polynomial function of h where E 2o Vk u is a holomorphic function of u in a neighbor Ns h vol V Vz dh JK ar nt t on N r N r k h u ve u 8 vol V V dh Nasr _ m A ET inne aiu Note that the function N is a polynomial function of degree N r whose homogeneous component of degree N r is the function Y h that is the volume of the polytope
54. rt contains At and has exactly r elements We now recall how to construct all maximal nested sets We may assume that AT is irreducible otherwise just take one of the irreducible components If M is a maximal nested set the vector space M AT is an hyperplane H thus an admissible hyperplane Definition 37 Let H be a At admissible hyperplane A maximal nested set M such that M At H is said attached to H Given M a MNS for At attached to H then M AT isa MNS for HN AT Therefore maximal nested sets for an irreducible set AT can be determined by induction over the set of A admissible hyperplanes For computing the Jeffrey Kirwan residue we only need some particular M N S s Let us briefly review the main ingredients Fix a total order ht on At Let M 5S S2 Sk be a set of subsets of At and choose in each S the element a maximal for the order given by ht This defines a map from M to At and we say that M is proper if O M M isa basis of V We denote by P A the set of MPNS So we have associated to every maximal proper nested set M an ordered basis gt by sorting the set M a1 a9 a of elements of AT Let v be an element in V not belonging to any admissible hyperplane Definition 38 Define P v A to be the set of M P A such that v C M C ai sae Sole When there is no possibility of confusion we will drop simply write P v for P v At We are now ready to state the b
55. ry simple example Observe that the root lattice is Ze Zez and vol V Vz dh 1 for the measure dh dhydhg There are two chambers namely c1 C e1 2 e1 and c2 C e1 e1 e2 Now let us compute the Jeffrey Kirwan residues on the chambers We have for example K 1 1 a J C1 a u2 1 JKey a 0 JK 1 1 1 4 a se 2 JK a D K 1 1 B J c1 a u2 JK gt 1 For the number of integral points we first note that F 0 0 1 2 1 2 Consequently Na h is equal to the Jeffrey Kirwan residue of f K 1 1 h plus fo K 1 2 1 2 h We rewrite the series fj j 1 2 as fj fi x eurhituzha Ju uy ug u1 u2 where Ge U1 7 ui ug x U1 U2 1 1 e7 1 e ui tu2 1 e7 u1 u2 U1 ui u2 ui U2 hi th SSS SSS ae ees eee ee 1 1 2 f Ipe n1 e itu 1 e u SED Using the series expansions 14 24 4527 O x and T sx O x we obtain that the number of integral points is the JK residue of 1 1 1 u1 u2 u1 u2 u u u2 ui ui u2 Omtptpoyuta u1 u2 u1 u2 l how 1 hathe ui ui u2 ui u2 u1 u2 u1 u2 F he h2 1 2 fi urlu u2 urlu u2 ur ur u2 ui u2 u1 u2 We then obtain doe as AT EIR al az oa air oa a erm Vhen are eae oe ath shat sph tM l h fhe ca Note that the functions Nat agree on walls that is hg 0 and the formulae above are valid o
56. s an open cone in a x ae such that F j is given by a polynomial formula p X u when A u R a we say that R is a domain of polynomiality and that p is the local polynomial for F on R Let us finally recall that the local polynomials p belong to some particular space of polynomials satisfying some system of partial differential equations For a a non compact root consider the derivative a We say that an hyperplane H t is admissible for A if H is spanned by a subset of dimt 1 non compact roots that is roots in A We denote by Hn the set of admissible hyperplanes for An Definition 11 A polynomial p on t is in the Dahmen Micchelli space D A if p satisfies the system of equations II a p 0 acAt Q for any Q Hp 16 Remark that the space D A depends only of A and not of a choice of A Then the following result follows from Dahmen Micchelli theory of the splines Proposition 12 For any domain of polynomiality R the polynomial u gt p A p belongs to the space D A 2 4 Quasi polynomiality results Let us come back to the discrete setting Let us fix as before a positive system of roots AT and consider the corresponding chambers a ae C t for At and At Fix Py Oa and u Pf Nae We can then define ma the multiplicity of 7 in the discrete series 7 By definition a quasipolynomial function on a lattice L is a function on L which coincides with a polynomial
57. t Yh DN Vz The domain D is a rational polyhedral cone which includes hfiz In practice this works only for small dimensions and when A is not too big We will program variations of these algorithms with less ambitious goals 36 3 7 Computation of Blattner formula general scheme In this subsection we summarize the steps to compute Blattner s formula and the general scheme to obtain the region of polynomiality The relative algorithms will be outlined in Section 6 Let G K T be given as in Section 2 2 Let A C t be the list of noncompact roots Our inputs are A Py and u Pf The goal is the study of the function u gt m Let At A A and recall that in this case a At admissible hyperplane is called a noncompact wall We use Blattner s formula In our notations 11 mh Y e w Nac we A pn wEWe 3 7 1 Numeric Input Py and p Pe Output a number The algorithm is clear e Compute AT A A and pp e Compute the Kostant function K G h for this system A e Compute a finite set F satisfying condition C e Compute a small element such that preg u does not belong to any affine hyperplane of the form wA H where H is a noncompact wall w We e Compute for all w W the number contributiony N4 wu pn using the algorithm described in 3 6 1 That is compute Ally P w reg A AT as explained in Fig 6 Then compute contributiony os el
58. tersburg Math J 4 1993 789 812 14 Hecht H and Schmid W A proof of Blattner s conjecture In vent Math 31 1975 129 154 15 Duflo M Heckman G and Vergne M Projection d orbites formule de Kirillov et formule de Blattner Harmonic analysis on Lie groups and symmetric spaces Kleebach 1983 M m Soc Math France N S 15 1984 65 28 16 Paradan P E Spin quantization and the K multiplicities of the dis crete series Ann Sci Ecole Norm Sup 4 36 2003 n 5 805 845 50
59. tion functions Let Vz be a lattice in V and suppose now that the elements a of our sequence At belong to the lattice Vz If h Vz we define N4 h T4 h NZ the number of integral points in the partition polytope IL4 h Thus N4 h is the number of solutions x1 2 y in non negative in tegers xj of the equation ys Lia h The function h N4 h is called the partition function of At We refer to it as Kostant partition function We will see after stating Theorem 24 that h gt N4 h is quasipolynomial on each chamber Let us recall briefly the theory that allows to compute Kostant partition func tions Jeffrey Kirwan residue Let v be a subset of 1 2 N We will say that v is generating respectively basic if the set a i v generates respectively is a basis of the vector space V We write Bases AT for the set of basic subsets Let R4 be the ring of rational functions on U the dual vector space to V with poles on hyperplanes determined by kernel of elements a AF 22 R a is Z graded by degree Every function in R 4 of degree r decomposes see 5 as the sum of basic fractions foe fo ear o Bases At and degenerate fractions here degenerate fractions are those for which the linear forms in the denominator do not span V Now having fixed a chamber c we define a functional JK f on R4 called the Jeffrey Kirwan residue or JK residue as follows vol a ifc c C a
60. xample 5 discrete 5 2 3 2 3 2 5 2 direction 1 0 0 1 gt function_discrete_mul_direction_lowest discrete 1 0 0 1 2 2 t 1 0 inf inf stands for oo Here the covering of N is N 0 00 NN and the polynomial is P t t 1 The output explicitly compute mpo t 1 tEN t gt 0 with u 7 2 3 2 3 2 7 2 the lowest K type of the representation 7 See Ex 8 for computation of the lowest K type Thus we compute the multiplicity starting from the lowest K type when we are moving off it in the direction of Vv In particular for t 100 we get m gt 008 101 as predicted in Ex 1 since discrete 5 2 3 2 3 2 5 2 and u 1000 is equal to Krep 207 2 3 2 3 2 207 2 Example 6 discrete 9 7 1 2 13 direction 1 0 0 0 1 gt function_discrete_mul_direction_lowest discrete direction 2 3 1 1 2 t 1 2 t 2 0 01 1 1 inf Here the covering is N 0 0 NN u 1 00 NN and the polynomials are P t 1 4t 4t on 0 0 NN and Py t 1 on 1 00 NN Observe that 0 0 NN 0 is just a point and that P 0 1 as it should be since p is the lowest K type Explicitly we simply compute mom 1 te N tO We conclude with one example in which the multiplicity grows the last poly nomial is not zero and has degree two We give more examples at the end of this article Example 7 discrete 57 2 39 2 3 2
61. ype of dimension d We verify that the multiplicities are bounded by d Ho133 27 2 9 2 7 2 5 2 3 2 5 2 lowHo133 15 6 5 1 0 4 bigHol33 15 1000 6 1000 5 1000 1 1000 1000 4 1000 verybigHol33 15 100000 6 10000 5 10000 1 10000 10000 4 100000 gt discretemult lowHol133 Ho133 3 3 1 TT 0 069 gt discretemult bigHol133 Hol133 3 3 4 TT 0 9714 gt discretemult verybigHol33 Hol33 3 3 4 TT 0 873 Fix now a K type p a direction v given by a dominant weight for K more details in Sec 3 7 3 and a discrete series parameter A The half line u tv stays inside the dominant chamber for K A very natural question is that of investigating the behavior of the multiplicity function as a function of t Z when we move from pu along the positive v direction that is the function t gt mi si t 2 O The answer to this question will be given by two sets of datas a covering of N determined by a finite number of closed intervals 7 C R with integral end points that is N Ui lt i lt s l NN together with polynomial functions P t 1 lt i lt s of degree bounded by pq p q 1 that compute the multiplicity on such intervals I in formula TH sg P t fr te LAN We remark two aspects First the 4 QN 1 lt i lt s constitutes a covering in the sense that we recover all of N but J A N A Jj41 A N can intersect in the extreme points and hence in this case Pj

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