Branching Rules for Principal Series Representations of SL(2) over a p-adic Field

Abstract

Abstract. We explicitly describe the decomposition into irreducibles of the restriction of the principal series representations of SL(2, k), for k a p-adic field, to each of its two maximal compact subgroups (up to conjugacy). We identify these irreducible subrepresentations in the Kirillov-type classification of Shalika. We go on to explicitly describe the decomposition of the reducible principal series of SL (2, k) in terms of the restrictions of its irreducible constituents to a maximal compact subgroup.

On branching rules of depth-zero representations

“…Proving results on branching rules at this level of generality is a new and novel step, and anticipates the development of a general theory out of the case-by-case analysis achieved to date. In this sense the current work complements a series by the author on branching rules of SL(2, k) [13,14,15] and, with P. Campbell, GL (3, k) [3,4]. Recently, U. Onn and P. Singla in [17] determined the complete decomposition into irreducible representations of the blocks of representations of [3].…”

On the Unicity of Types for Toral Supercuspidal Representations

“…In SL 2 (F ), there are two conjugacy classes of unramified anisotropic tori, attached to the distinct conjugacy classes of vertices in B(SL 2 , F ). There are between 2 and 4 conjugacy classes of ramified anisotropic tori, attached to the midpoint of facets; see [18], for example. With respect to the coordinates above, the roots of each SL 2 (F ) subgroup are ±2ε i , and thus up to conjugacy we can arrange that the vertices have e i -coordinates in {0, 1 2 }, whereas the midpoints have e i -coordinate 1 4 .…”

Branching Rules for n-fold Covering Groups of SL2 over a Non-Archimedean Local Field

“…The set S is a lift to the covering group K 1 of a similar set of double coset representatives calculated in [16]. Using the latter set, and because µ n ⊂ B 1 l , it is easy to see that…”

“…Our method is aligned with the one in [16] for the linear group SL 2 (F); however, the technicalities in the covering case are much more involved than the linear case, and the results are fairly different. For instance, the K-type decomposition is no longer multiplicity-free (Corollary 2).…”