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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

##### MSC:
 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