REPRESENTATION THEORY OF THE ®-DETERMINANT AND ZONAL SPHERICAL FUNCTIONS KAZUFUMI KIMOTO Abstract. U ¢ (®) l We investigate the structure of the cyclic module Vn;l(®) = (gln) det (X) by embedding it to the tensor product space (Cn)­nl and utilizing the Schur-Weyl duality. We show that the entries of the ¸ transition matrices Fn;l(®) are given by a variation of the spherical Fourier transformation of a certain class n function on Snl with respect to the subgroup Sl (Theorem 1.4). This result also provides another proof of (2l¡s;s) Theorem ??. Further, we calculate the polynomial F2;l (®) by using an explicit formula of the values of zonal spherical functions for the Gelfand pair (S2l; Sl £ Sl) due to Bannai and Ito (Theorem 2.1). 1. Irreducible decomposition of Vn;l(®) and transition matrices Let us ¯x n; l 2 N. Consider the standard tableau T with shape (ln) such that the (i; j)-entry of T is (i ¡ 1)l + j. For instance, if n = 3 and l = 2, then 1 2 T = 3 4 : 5 6 We denote by K = R(T) and H = C(T) the row group and column group of the standard tableau T respectively. Namely, © ¯ ª (1.1) K = g 2 S ¯ dg(x)=le = dx=le ; x 2 [nl] ; © nl ¯ ª ¯ (1.2) H = g 2 Snl g(x) ´ x (mod l); x 2 [nl] : We put 1 X (1.3) e = k 2 C[S ]: jKj nl k2K This is clearly an idempotent element in C[S ]. Let ' be a class function on H. We put nl X © = '(h)h 2 C[Snl]: h2H Cn ­nl U C Consider the tensor product space V = ( ) . We introduce a ( (gln); [Snl])-module structure of V by Xnl ¢ ­ ¢ ¢ ¢ ­ ­ ¢ ¢ ¢ ­ s-th ­ ¢ ¢ ¢ ­ Eij ei1 einl = ±is;j ei1 ei einl ; s=1 ­ ¢ ¢ ¢ ­ ¢ ­ ¢ ¢ ¢ ­ 2 ei1 einl σ = eiσ(1) eiσ(nl) (σ Snl); f gn Cn where ei i=1 denotes the standard basis of . The main concern of this subsection is to describe the irreducible U ¢ decomposition of the left (gln)-module V e©e. We ¯rst show that Vn;l(®) is isomorphic to V ¢ e©e for a special choice of '. Consider the group isomorphism ! l θ : H Sn de¯ned by θ(h) = (θ(h)1; : : : ; θ(h)l); θ(h)i(x) = y () h((x ¡ 1)l + i) = (y ¡ 1)l + i: Date: October 30, 2008. 2000 Mathematics Subject Classi¯cation. Primary 22E47, 33C45; Secondary 43A90, 13A50. Key words and phrases. Alpha-determinant, cyclic modules, Jacobi polynomials, singly confluent Heun ODE, permanent, Kostka numbers, irreducible decomposition, spherical Fourier transformation, zonal spherical functions, Gelfand pair. 1 2 KAZUFUMI KIMOTO We also de¯ne an element D(X; ') 2 A(Matn) by X Yn Yl X Yn Yl D(X; ') = '(h) x = '(h) x ¡1 θ(h)p(q);q q,θ(h)p (q) h2H q=1 p=1 h2H q=1 p=1 X Yn Yl ¡1 = '(θ (σ1; : : : ; σl)) xσp(q);q: σ1,...,σl2Sn q=1 p=1 º(¢) (®) l ¡1 ¢ ¢ ¢ 2 n We note that D(X; ® ) = det (X) since ºθ (σ1; : : : ; σl) = ºσ1 + + ºσl for (σ1; : : : ; σl) Sl . Take a class function ± on H de¯ned by H ( 1 h = 1 ±H (h) = 0 h =6 1: l We see that D(X; ±H ) = (x11x22 : : : xnn) . We need the following lemma (The assertion (1) is just a rewrite of Lemma ??, and (2) is immediate to verify). Lemma 1.1. (1) It holds that U ¢ ­l ­ ¢ ¢ ¢ ­ ­l ¢ Sl Cn ­n (gln) e1 en = V e = ( ) ; M Yn Yl U ¢ C ¢ » Sl Cn ­n (gln) D(X; ±H ) = xipq q = ( ) : ipq 2f1;2;:::;ng q=1 p=1 (1·p·l; 1·q·n) (2) The map Yn Yl T U ¢ 3 : (gln) D(X; ±H ) xipq q q=1 p=1 7¡! ­ ¢ ¢ ¢ ­ ­ ¢ ¢ ¢ ­ ­ ¢ ¢ ¢ ­ ¢ 2 ¢ (ei11 eil1 ) (ei1n eiln ) e V e U ¤ is a bijective (gln)-intertwiner. We see that T (D(X; ')) à ! X Yn Yl T = '(h) xθ(h)p(q);q 2 hXH q=1 p=1 ­ ¢ ¢ ¢ ­ ­ ¢ ¢ ¢ ­ ­ ¢ ¢ ¢ ­ ¢ = '(h)(eθ(h)1(1) eθ(h)l(1)) (eθ(h)1(n) eθ(h)l(n)) e 2 h H X ­l ­ ¢ ¢ ¢ ­ ­l ¢ ¢ ­l ­ ¢ ¢ ¢ ­ ­l ¢ = e1 en '(h)h e = e1 en e©e h2H by (2) in Lemma 1.1. Using (1) in Lemma 1.1, we have the Lemma 1.2. It holds that U ¢ » ¢ (gln) D(X; ') = V e©e U ¢ » º(h) ¤ as a left (gln)-module. In particular, V e©e = Vn;l(®) if '(h) = ® . By the Schur-Weyl duality, we have M » M¸ £ S¸ V = n : ¸`nl ¸ Here S denotes the irreducible unitary right Snl-module corresponding to ¸. We see that ¡ ¢ D E S¸ ¢ G S¸ dim e = indK 1K ; = K¸(ln); Snl REPRESENTATION THEORY OF THE ®-DETERMINANT AND ZONAL SPHERICAL FUNCTIONS 3 where 1 is the trivial representation of K and h¼; ½i is the intertwining number of given representations ¼ K Snl and ½ of Snl. Since K¸(ln) = 0 unless `(¸) · n, it follows the Theorem 1.3. It holds that M ¡ ¢ ¢ » M¸ £ S¸ ¢ V e©e = n e©e : ¸`nl `(¸)·n In particular, as a left U(gl )-module, the multiplicity of M¸ in V ¢ e©e is given by n ¡ ¢ n ¸ dim S ¢ e©e = rkEnd(S¸¢e)(e©e): ¤ ` · f ¸ ¸ g Let ¸ nl be a partition such that `(¸) n and put d = K¸(ln). We ¯x an orthonormal basis e1 ;:::; ef ¸ S¸ ¸ ¸ S¸ K of such that the ¯rst d vectors e1 ;:::; ed form a subspace ( ) consisting of K-invariant vectors and left ¸ ¸ K f ¡d vectors form the orthocomplement of (S ) with respect to the Snl-invariant inner product. The matrix coe±cient of S¸ relative to this basis is ­ ® ¸ ¸ ¢ ¸ 2 · · ¸ (1.4) Ãij(g) = ei g; ej S¸ (g Snl; 1 i; j f ): M¸ ¢ We notice that this function is K-biinvariant. We see that the multiplicity of n in V e©e is given by the rank of the matrix à ! X ¸ '(h)Ãij(h) : h2H 1·i;j·d As a particular case, we obtain the M¸ U ¢ (®) l Theorem 1.4. The multiplicity of the irreducible representation n in the cyclic module (gln) det (X) is equal to the rank of à ! X ¸ º(h) ¸ (1.5) Fn;l(®) = ® Ãij(h) ; h2H 1·i;j·d f ¸ g n where Ãij i;j denotes a basis of the ¸-component of the space C(K Snl=K) of K-biinvariant functions on Snl given by (1.4). ¤ ¸ f ¸ g Remark 1.5. (1) We have Fn;l(0) = I by the de¯nition of the basis Ãij i;j in (1.4). º(g¡1) º(g) ¸ ¡1 ¸ 2 ¸ ¤ (2) Since ® = ® and Ãij(g ) = Ãji(g) for any g Snl, the transition matrices satisfy Fn;l(®) = ¸ Fn;l(®). (3) In Examples 1.6 and 1.8 below, the transition matrices are given by diagonal matrices. We expect that ¸ C any transition matrix Fn;l(®) is diagonalizable in MatK¸(ln) ( [®]). Example 1.6. If l = 1, then H = G = Sn and K = f1g. Therefore, for any ¸ ` n, we have ­ ® ¸ n! ¸ (1.6) Fn;1(') = ';  I f ¸ Sn ¸ by the orthogonality of the matrix coe±cients. Here  denotes the irreducible character of Sn corresponding to ¸. In particular, if ' = ®º(¢), then ¸ (1.7) Fn;1(®) = f¸(®)I º(¢) since the Fourier expansion of ® (as a class function on Sn) is X f ¸ (1.8) ®º(¢) = f (®)¸; n! ¸ ¸`n which is obtained by specializing the Frobenius character formula for Sn (see, e.g. [10]). 4 KAZUFUMI KIMOTO (nl) S(nl) Example 1.7. Let us calculate Fn;l (®) by using Theorem 1.4. Since is the trivial representation, it follows that (S(nl))K = S(nl) and X X (nl) º(h) h ¢ i º(σ1) º(σl) Fn;l (®) = ® e h; e = ® : : : ® h2H σ1,...,σl2Sn = ((1 + ®)(1 + 2®) ::: (1 + (n ¡ 1)®))l ; where e denotes a unit vector in S(nl). (nl¡1;1) Example 1.8. Let us calculate Fn;l (®) by using Theorem 1.4. As is well known, the irreducible (right) (nl¡1;1) nl Snl-module S can be realized in C as follows: ( ¯ ) ¯ ¯ Xnl S(nl¡1;1) nl 2 Cnl ¯ = (xj)j=1 ¯ xj = 0 : j=1 This is a unitary representation with respect to the ordinary hermitian inner product h¢; ¢i on Cnl. It is immediate to see that ¡ ¢ S(nl¡1;1) K n ¯ o ¯ nl 2 S(nl¡1;1) ¢ ¢ ¢ · = (xj)j=1 ¯ xpl+1 = xpl+2 = = x(p+1)l (0 p < n) : ¡ ¢ (nl¡1;1) K Take an orthonormal basis e1;:::; en¡1 of S by l l l ¡z }| { z }| { z }| {¢ 1 j j 2j 2j nj nj ej = p ! ;:::;! ; ! ;:::;! ;:::; ! ;:::;! (1 · j · n ¡ 1); nl (nl¡1;1) where ! is a primitive n-th root of unity. Then, the (i; j)-entry of the transition matrix Fn;l (®) is X X Xn Xl 1 ¡ ®º(h) he ¢ h; e i = ®º(σ1) : : : ®º(σl)!σq (p)i pj i j nl h2H σ1,...,σl2Sn p=1 q=1 à !l¡1 à ! X 1 X Xn = ®º(¿) ®º(σ)!σ(p)i¡pj : n ¿2Sn σ2Sn p=1 ¡ The ¯rst factor is ((1 + ®)(1 + 2®) ::: (1 + (n ¡ 1)®))l 1. We show that 1 X Xn ®º(σ)!σ(p)i¡pj = (1 ¡ ®)(1 + ®)(1 + 2®) ::: (1 + (n ¡ 2)®)± n ij σ2Sn p=1 (i; j = 1; 2; : : : ; n ¡ 1): For this purpose, by comparing the coe±cients of ®n¡m in both sides, it is enough to prove ½· ¸ · ¸¾ 1 X Xn n ¡ 1 n ¡ 1 !σ(p)i¡pj = ¡ ± n m ¡ 1 m ij σ2Sn p=1 º(σ)=n¡m (i; j; m = 1; 2; : : : ; n ¡ 1); £ ¤ n where m denotes the Stirling number of the ¯rst kind (see, e.g.