The Double Cover of Odd General Spin Groups, Small Representations, and Applications

Abstract

Abstract. We construct local and global metaplectic double covers of odd general spin groups, using the cover of Matsumoto of spin groups. Following Kazhdan and Patterson, a local exceptional representation is the unique irreducible quotient of a principal series representation, induced from a certain exceptional character. The global exceptional representation is obtained as the multi-residue of an Eisenstein series, it is an automorphic representation and decomposes as the restricted tensor product of local exceptional representations. As in the case of the small representation of SO 2n+1 of Bump, Friedberg and Ginzburg, exceptional representations enjoy the vanishing of a large class of twisted Jacquet modules (locally), or Fourier coefficients (globally). Consequently they are useful in many settings, including lifting problems and Rankin-Selberg integrals. We describe one application, to a calculation of a co-period integral.Let G n = GSpin 2n+1 be the split odd general spin group of rank n + 1. Its derived group is G ′ n = Spin 2n+1 , the simple split simply-connected algebraic group of type B n . The group G n occurs as a Levi subgroup of G ′ n+1 . For a local field F of characteristic 0, letG ′ n+1 (F ) be the metaplectic double cover of G ′ n+1 (F ) defined by Matsumoto [58]. We can obtain a double coverG n (F ) of G n (F ) by restriction.Following Banks, Levi and Sepanski [10] we define a section s and a 2-cocycle σ of G ′ n+1 (F ), representing the cohomology class in H 2 (G ′ n+1 (F ), {±1}) ofG ′ n+1 (F ). We show that the restriction of σ to G n (F ) × G n (F ) satisfies a block-compatibility relation, with respect to standard Levi subgroups. This is a useful condition for studying parabolically induced representations.Fix a Borel subgroup in G n (F ). The preimageT n+1 (F ) of the maximal torus T n+1 (F ) in the cover is a two step nilpotent subgroup, its irreducible genuine representations are parameterized by genuine characters of its center CT n+1 (F ) . The analogous theory for covers of GL n was developed by Kazhdan and Patterson [48], who studied a special class of genuine characters which they called "exceptional". In our setting these are characters χ of CT n+1 (F ) satisfying χ(α ∨ * (x l(α) )) = x for all simple roots α of G n and x ∈ F * , where α ∨ * is a certain lift of the coroot α ∨ to the cover and l(α) is the length of α.We use an exceptional character χ to construct a genuine principal series representation ofG n (F ), which has a unique irreducible quotient denoted Θ = Θ Gn,χ . The quotient Θ is an exceptional representation, or a small representation in the terminology of Bump, Friedberg and Ginzburg [16]. Our purpose is to develop a theory of these representations.Let O be a unipotent class of G n , it corresponds to a partition of 2n + 1 for which an even number appears with an even multiplicity. Let V O be the corresponding unipotent subgroup. We consider certain characters of V O (F ), called "generic". Roughly, a character ψ of V O (F ) is generic if it is in general p...

Theta distinguished representations, inflation and the symmetric square L-function

“…Here Θ and Θ belong to the exceptional, or small, representation of Bump et al [7], whose analog for G n was described in [41]. This integral was studied in [40] for SO 2n+1 (extended in [41] to G n ) and we proved the implication (2) ⇒ (3).…”

“…This integral was studied in [40] for SO 2n+1 (extended in [41] to G n ) and we proved the implication (2) ⇒ (3). In the setting of G n , one can also study the twisted symmetric square L-function.…”

“…The theory of Bump et al [7] can be extended to G n (F), mainly because both groups are of type B n and in particular, the unipotent subgroups are isomorphic. The details were carried out in [41]. Our results here are stated for G n , but apply similarly to SO 2n+1 and θ SO 2n+1 (see Sect.…”

“…The proof of Theorem 1.1 is essentially the local analog of the global computation of the co-period in [40,41]. Write Q n = M n U n and let C = C U n be the center of the unipotent radical U n .…”

“…This isomorphism is not canonical. We describe the choice used in [41], which is convenient for certain computations (see below).…”