Restriction of square integrable representations: discrete spectrum
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 \textit{R. Hotta} and \textit{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\((2k)\times \text{Spin}(2n-2k,1)\).
scientific article; zbMATH DE number 2103767
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