## Restriction of square integrable representations: discrete spectrum.(English)Zbl 1056.22008

Let $$G$$ be a semisimple Lie group and let $$H$$ be a reductive subgroup invariant under the Cartan involution. Let $$K$$ be a maximal compact subgroup of $$G$$. Hence we have $${\mathfrak g}={\mathfrak k}\oplus {\mathfrak p}={\mathfrak h}\oplus{\mathfrak q}$$. Let $$V$$ be a discrete series representation with Harish-Chandra parameter $$\lambda$$ and let $$(\tau, W)$$ be its lowest $$K$$-type. Following R. Hotta and R. Parthasarathy [Invent. Math. 26, 133–178 (1974; Zbl 0298.22013)], one realizes $$V$$ as the corresponding eigenvectors of the Casimir element acting on the $$L^2$$-sections of the $$G$$-equivalent homogeneous bundle $$G\times_\tau W\to G/K$$. Hence $$V$$ consists of real analytic functions and it makes sense to restrict $$V$$ to $$H\times_\tau W\to H/(H\cap K)$$. We will denote the restriction by $$r$$ and the $$L^2$$ sections of $$H\times_\tau W$$ by $$L^2(H,\tau_*)$$. The main theorem of this paper shows that the image of $$r$$ lies in $$L^2(H,\tau_*)$$ and $$r:V\to L^2(H,\tau_*)$$ is continuous. Let $$r^*$$ denote the adjoint operator of $$r$$. The authors show that the Berezin transform $$rr^*$$ is a continuous linear operator on $$L^2(H,\tau_*)$$. The above is generalized in Theorem 3 of the paper: We consider $$\tau_m=\text{Sym}^m({\mathfrak p}\cap{\mathfrak q})\otimes W$$ as an $$H\cap K$$-module and we set $$L_m:= L^2(H,\tau_m)$$. There exists a continuous $$H$$-linear map $$\tau_m:V\to L_m$$ such that $$\bigoplus_mr_m:V\to \bigoplus_mL_m$$ is injective. This result overlaps with the works of Kobayashi, Loke, Martens, and Gross and Wallach in which they consider the restrictions of Harish-Chandra modules of $$G$$ which are $$H\cap K$$-admissible. In the last section, the authors study the restriction of principal series representations of Spin$$(2n,1)$$ to Spin$$(2k)\times \text{Spin}(2n-2k,1)$$.

### MSC:

 22E46 Semisimple Lie groups and their representations 43A85 Harmonic analysis on homogeneous spaces

### Keywords:

Berezin transform; discrete series representation

Zbl 0298.22013
Full Text:

### References:

 [1] M. Atiyah and W. Schmid, A geometric construction of the discrete series for semisimple Lie groups , Invent. Math. 42 (1977), 1–62. · Zbl 0373.22001 [2] E. P. van den Ban, Invariant differential operators on a semisimple symmetric space and finite multiplicities in a Plancherel formula , Ark. Mat. 25 (1985), 175–187. · Zbl 0645.43009 [3] L. Bers and M. Schechter, “Elliptic equations” in Partial Differential Equations (Boulder, Colo., 1957) , Lectures in Appl. Math. 3 , Interscience, New York, 1964, 131–299. [4] M. Cowling, The Kunze-Stein phenomenon , Ann. of Math. (2) 107 (1978), 209–234. JSTOR: · Zbl 0363.22007 [5] B. Gross and N. Wallach, “Restriction of small discrete series representations to symmetric subgroups” in The Mathematical Legacy of Harish-Chandra (Baltimore, 1998) , ed. R. S. Doran and V. S. Varadarajan, Proc. Sympos. Pure Math. 68 , Amer. Math. Soc., Providence, 2000, 255–272. · Zbl 0960.22008 [6] Harish-Chandra, “Some results on differential equations” in Collected Papers, Vol. III ( 1959–1968.) , ed. V. S. Varadarajan, Springer, New York, 1984, 4–48. · Zbl 0161.33803 [7] H. Hecht and W. Schmid, A proof of Blattner’s conjecture , Invent. Math. 31 (1975), 129–154. · Zbl 0319.22012 [8] R. Hotta and R. Parthasarathy, Multiplicity formulae for discrete series , Invent. Math. 26 (1974), 133–178. · Zbl 0298.22013 [9] R. Howe, “Reciprocity laws in the theory of dual pairs” in Representation Theory of Reductive Groups (Park City, Utah, 1982) , ed. P. C. Trombi, Progr. Math. 40 , Birkhäuser, Boston, 1983, 156–175. · Zbl 0543.22009 [10] H. P. Jacobsen and M. Vergne, Restrictions and expansions of holomorphic representations , J. Funct. Anal. 34 (1979), 29–53. · Zbl 0433.22011 [11] A. Knapp, Representation Theory of Semisimple Lie Groups: An Overview Based on Examples , Princeton Math. Ser. 36 , Princeton Univ. Press, Princeton, 1986. · Zbl 0604.22001 [12] ——–, Lie Groups beyond an Introduction , Progr. Math. 140 , Birkhäuser, Boston, 1996. · Zbl 0862.22006 [13] T. Kobayashi, The restricton of $$A_\mathfrak q(\lambda)$$ to reductive subgroups , Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), 262–267. · Zbl 0826.22014 [14] –. –. –. –., Discrete decomposability of the restriction of $$A_\mathfrak q(\lambda)$$ with respect to reductive subgroups and its applications , Invent. Math. 117 (1994), 181–205. · Zbl 0826.22015 [15] –. –. –. –., Discrete decomposability of the restriction of $$A_\mathfrak q(\lambda)$$ with respect to reductive subgroups, II: Micro-local analysis and asymptotic $$K$$-support , Ann. of Math. (2) 147 (1998), 709–729. JSTOR: · Zbl 0910.22016 [16] –. –. –. –., Discrete decomposability of the restriction of $$A_\mathfrak q(\lambda)$$ with respect to reductive subgroups, III: Restriction of Harish-Chandra modules and associated varieties , Invent. Math. 131 (1998), 229–256. · Zbl 0907.22016 [17] –. –. –. –., Discrete series representations for the orbit spaces arising from two involutions of real reductive Lie groups , J. Funct. Anal. 152 (1998), 100–135. · Zbl 0937.22008 [18] –. –. –. –., “Harmonic analysis on homogeneous manifolds of reductive type and unitary representation theory” in Selected Papers on Harmonic Analysis: Groups and Invariants , ed. K. Nomizu, Amer. Math. Soc. Transl. Ser. 2 183 , Amer. Math. Soc., Providence, 1998, 1–31. · Zbl 0895.22006 [19] –. –. –. –., “Discretely decomposable restrictions of unitary representations of reductive Lie groups –.-examples and conjectures” in Analysis on Homogeneous Spaces and Representation Theory of Lie Groups , ed. T. Kobayashi, Adv. Stud. Pure Math. 26 , Math. Soc. Japan, Tokyo, 2000, 99–127. · Zbl 0959.22009 [20] H. Y. Loke, Restrictions of quaternionic representations , J. Funct. Anal. 172 (2000), 377–403. · Zbl 0953.22018 [21] G. W. Mackey, The Theory of Unitary Group Representations , Chicago Lectures in Math., Univ. of Chicago Press, Chicago, 1967. · Zbl 0162.09401 [22] S. Martens, The characters of the holomorphic discrete series , Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 3275–3276. · Zbl 0308.22013 [23] G. Ólafsson, Symmetric spaces of Hermitian type , Differential Geom. Appl. 1 (1991), 195–233. · Zbl 0785.22021 [24] G. Ólafsson and B. \Orsted, The holomorphic discrete series for affine symmetric spaces, I , J. Funct. Anal. 81 (1988), 126–159. · Zbl 0678.22008 [25] R. Takahashi, Sur les représentations unitaires des groupes de Lorentz généralisés , Bull. Soc. Math. France 91 (1963), 289–433. · Zbl 0196.15501 [26] F. Trèves, Topological Vector Spaces, Distributions and Kernels , Academic Press, New York, 1967. · Zbl 0171.10402 [27] P. C. Trombi and V. S. Varadarajan, Asymptotic behaviour of eigen functions on a semisimple Lie group: The discrete spectrum , Acta Math. 129 (1972), 237–280. · Zbl 0244.43006 [28] J. A. Vargas, Restriction of some discrete series representations , Algebras Groups Geom. 18 (2001), 85–100. · Zbl 1008.22005 [29] –. –. –. –., Restriction of holomorphic discrete series to real forms , Rend. Sem. Mat. Univ. Politec. Torino 60 (2002), 45–53. · Zbl 1177.22009 [30] N. R. Wallach, On the Enright-Varadarajan modules: A construction of the discrete series , Ann. Sci. École Norm. Sup. (4) 9 (1996), 81–101. · Zbl 0379.22008 [31] N. R. Wallach and J. A. Wolf, “Completeness of Poincare series for automorphic forms associated to the integrable discrete series” in Representation Theory of Reductive Groups (Park City, Utah, 1982) , ed. P. C. Trombi, Progr. Math. 40 , Birkhäuser, Boston, 1983, 265–281. · Zbl 0566.22014 [32] G. Warner, Harmonic Analysis on Semi-Simple Lie Groups, I, II , Grundlehren Math. Wiss. 188 , 189 , Springer, New York, 1972. Mathematical Reviews (MathSciNet): · Zbl 0265.22020 [33] J. A. Wolf, “Representations that remain irreducible on parabolic subgroups” in Differential Geometrical Methods in Mathematical Physics (Aix-en-Provence and Salamanca, 1979) , Lecture Notes in Math. 836 , Springer, Berlin, 1980, 129–144. [34] G. Zhang, Berezin transform on real bounded symmetric domains , Trans. Amer. Math. Soc. 353 (2001), 3769–3787. JSTOR: · Zbl 0965.22015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.