Notes on Stein-Sahi representations and some problems of non-L 2 harmonic analysis

Abstract

We discuss one natural class of kernels on pseudo-Riemannian symmetric spaces.Recently, Oshima [67] published his formula for c-function for L 2 on pseudo-Riemannian symmetric spaces (see also works of Delorm [12] and van den Ban-Schlichtkrull [3], [4]). After this, there arises a natural question about other solvable problems of non-commutative harmonic analysis.In the Appendix to the paper [57], the author proposed a series of non L 2inner products in spaces of functions on pseudo-Riemannian symmetric spaces and conjectured that this object is reasonable and admits an explicit harmonic analysis.In this work, we discuus the problem in more details, in particular, we obtain the Plancherel formula for these kernel for Riemannian symmetric spaces U(n), U(n)/O(n), U(2n)/Sp(n). We also give a new proof of Sahi's results [76].our inner product is non-negative definite.The latter statement is a reformulation of the well-known theorem (Berezin-Gindikin-Rossi-Vergne-Wallach) on unitarizability of scalar highest weight representations ([6], [83], see a relatively recent exposition in [18]).1 The first work that can be attributed to this subject, is Vershik, Gelfand, Graev,[84], [20]. These authors apply the representations ρσ of the groups U(1, q) to construct representations of current groups; for a collection of other early references, see [56].2 Compare the Stein inner product (0.6) and the Berezin inner product (0.11). In the first case, the kernel has a singularity on the surface det(x − y) = 0. This surface itself is singular, and its most singular strate is the diagonal x = y. On the contary, the Berezin kernel has no singularities in Bp,q × Bp,q at all. We can substitute to (0.11) arbitrary compactly supported distributions unstead of functions F 1 , F 2 .

Stein–Sahi Complementary Series and Their Degenerations

“…In a certain sense, the integral (3.21) is a multivariate analog of the Lobachevsky integral. On the other hand, (3.21) is a special case of the modified Kadell integral [29].…”

“…The only difficulty is Theorem 3.11 (the expansion of the integral kernel in characters, we choose G = U(n, n), because this can be done by elementary tools). In the general Hermitian case, one can refer to the version of the Kadell integral [20] from [29] (the integrand is a product of a Jack polynomial and a Selberg-type factor.…”

Restriction of Representations of GL (n + 1, ℂ) to GL (n, ℂ) and Action of the Lie Overalgebra

“…Also, some explicitly solvable spectral problems in representation theory can be regarded as special cases of the restriction problem. In particular, a decomposition of L 2 on a classical pseudo-Riemannian symmetric space G/H can be regarded as a special case of the restriction of a Stein type principal series of a certain overgroup G to the symmetric subgroup G,[21]. So, for L 2 on symmetric spaces the problem of action of the overalgebra discussed below makes sense.3 On Sturm-Liouville in imaginary direction, see[23] and further references in that paper, see also[10].…”