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Analytic representations of \(SL_ 2\) over a \({\mathfrak p}\)-adic number field. II. (English) Zbl 0549.22008
Automorphic forms of several variables, Taniguchi Symp., Katata/Jap. 1983, Prog. Math. 46, 282-297 (1984).
[For the entire collection see Zbl 0534.00006].
Let L be a finite extension of \({\mathbb{Q}}_ p\) and set \(G=SL(2,L)\). In an earlier joint work with A. Murase [J. Fac. Sci., Univ. Tokyo, Sect. I A 28, 891-905 (1981; Zbl 0498.22016)] the author studied a p-adic analogue for G of the holomorphic discrete series representations of SL(2,\({\mathbb{R}})\); these are representations of G acting on a \({\mathbb{Q}}_ p\)-vector space of appropriately defined ”analytic” functions (with values in an algebraically closed field k containing L). In the present paper, the author relates these ”holomorphic discrete series representations” to a family of principal series, the results being analogous to the familiar theory for SL(2,\({\mathbb{R}})\); irreducibility and equivalence results for the discrete series - only conjectured in the earlier work just alluded to - are obtained by way of similar results for the induced (principal) series representations. We emphasize that the induced representations - attached as they are to characters \(\chi\) : \(L^ x\to k^ x\), are realized on spaces of k-valued functions, i.e., vector spaces over \({\mathbb{Q}}_ p\) rather than \({\mathbb{C}}\) or \({\mathbb{Q}}_{\ell}\), \(\ell\neq p\). Where these intriguing results will lead is not yet clear.
Reviewer: S.Gelbart

22E50 Representations of Lie and linear algebraic groups over local fields
22E35 Analysis on \(p\)-adic Lie groups
20G05 Representation theory for linear algebraic groups