Let E be a nonarchimedean local field of characteristic zero and residual characteristic p. Let G be a connected reductive group defined over E and π an irreducible admissible representation of G(E). A result of C. Moeglin and J.-L. Waldspurger (for p = 2) and S. Varma (for p = 2) states that the leading coefficient in the character expansion of π at the identity element of G(E) gives the dimension of a certain space of degenerate Whittaker forms. In this paper we generalize this result of Moeglin and Waldspurger to the setting of covering groups of G(E).

In this note we prove a certain multiplicity formula regarding the restriction of an irreducible admissible genuine representation of a 2-fold cover GL 2 (F ) of GL 2 (F ) to the 2-fold cover SL 2 (F ) of SL 2 (F ), and find in particular that this multiplicity may not be one, a result that was recently observed for certain principal series representations in the work of Szpruch (2013). The proofs follow the standard path via Waldspurger's analysis of theta correspondence between SL 2 (F ) and PGL 2 (F ).

Abstract. Let E/F be a quadratic extension of a non-Archimedian local field. Splitting of the 2-fold metaplectic cover of Sp 2n (F) when restricted to various subgroups of Sp 2n (F) plays an important role in application of the Weil representation of the metaplectic group. In this paper we prove the splitting of the metaplectic cover of GL 2 (E) over the subgroups GL 2 (F) and D × F , where D F is the quaternion division algebra with center F, as a first step in our study of the restriction of representations of metaplectic cover of GL 2 (E) to GL 2 (F) and D × F . These results were suggested to the author by Professor Dipendra Prasad.

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Let G \mathbf {G} be the General Linear or Special Linear group with entries from the finite quotients of the ring of integers of a non-archimedean local field and U \mathbf {U} be the subgroup of G \mathbf {G} consisting of upper triangular unipotent matrices. We prove that the induced representation Ind U G ⁡ ( θ ) \operatorname {Ind}^{\mathbf {G}}_{\mathbf {U}}(\theta ) of G \mathbf {G} obtained from a non-degenerate character θ \theta of U \mathbf {U} is multiplicity free for all ℓ ≥ 2. \ell \geq 2. This is analogous to the multiplicity one theorem regarding Gelfand-Graev representation for the finite Chevalley groups. We prove that for many cases the regular representations of G \mathbf {G} are characterized by the property that these are the constituents of the induced representation Ind U G ⁡ ( θ ) \operatorname {Ind}^{\mathbf {G}}_{\mathbf {U}}(\theta ) for some non-degenerate character θ \theta of U \mathbf {U} . We use this to prove that the restriction of a regular representation of General Linear groups to the Special Linear groups is multiplicity free and also obtain the corresponding branching rules in many cases.