International Congress of Mathematicians

618 T. Kobayashito Kirillov-Kostant. This representation is a unitarization of a Zuckerman-Voganmodule A q (A) after some p-shift, and can be realized in the Dolbeault cohomologygroup on Ox by the results of Schmid and Wong. (Here, we adopt the samepolarization and normalization as in a survey [K04, §2], for the geometric quantizationOx =$• n\.) We note that nx £ G for "most" A. Let g = 6 + p be thecomplexification of a Cartan decomposition of the Lie algebra go of G. We setA+(p) — {a £ A(p,t) : (A, a) > 0}, for A G V^ïl* 0 -The original proof (see [K05]) of the next theorem was based on an algebraic methodwithout using microlocal analysis. Theorem B gives a simple and alternative proof.Theorem C ([K05]). Letnx £ G be attached to an elliptic coadjoint orbit Ox. Ifthen the restriction nx\a' is G'-admissible.R-span A+(p) n Cone(G') = {0}, (2.2)Let us illustrate Theorem C in Examples 2.6 and 2.7 for non-compact G'.For this, we note that a maximal compact subgroup K is sometimes of the formKi x K 2 (locally). This is the case if G/K is a Hermitian symmetric space (e.g.G = Sp(n,R),SO* (2n), SU(p, qj). It is also the case if G = 0(p,q), Sp(p,q), etc.Example 2.6 (K ~ Ki x K 2 ). Suppose K is (locally) isomorphic to the directproduct group Ki x K 2 . Then, tie restriction nx\a' is G'-admissible if A| tn e 2 = 0and G' D Ki. So does the restriction n\a' if n is any subquotient of a coherentcontinuation of nx- This case was a prototype of G'-admissible restrictions n\a'(where G' is non-compact and n is a non-highest weight module) proved in 1989 bythe author ([K01; K02, Proposition 4.1.3]), and was later generalized to Theorems Band C. Special cases include:(1) Ki ~ T, then n is a unitary highest weight module. The admissibility ofthe restrictions n\a' in this case had been already known in '70s (see Alartens[Alt], Jakobson-Vergne [J-Vr]).(2) Ki ~ SU(2), then nx is a quaternionic discrete series. Admissible restrictionsn\(}i in this case are especially studied by Gross and Wallach [Gr-Wi] in '90s.(3) Ki ~ 0(q),U(q),Sp(q). Explicit branching laws of the restriction nx\a' forsingular A are given in [K03, Part I] with respect to the vertical inclusions ofthe diagram below (see also [Koi,Kos] for those to horizontal inclusions).0(4p,4q) D U(2p,2q) D Sp(p,q)U U U0(4r) x 0(4p - Ar, Aq) D U(2r) x U(2p - 2r, 2q) D Sp(r) x Sp(p - r, q)