Metaplectic Cover Research Articles

Functional equations and gamma factors of local zeta functions for the metaplectic cover of SL2

We introduce a local zeta-function for an irreducible admissible supercuspidal representation π of the metaplectic double cover of SL2 over a non-archimedean local field of characteristic zero. We prove a functional equation of the local zeta-functions showing that the gamma factor is given by a Mellin type transform of the Bessel function of π. We obtain an expression of the gamma factor, which shows its entireness on C. Moreover, we show that, through the local theta-correspondence, the local zeta-function on the covering group is essentially identified with the local zeta-integral for spherical functions on PGL2≅SO3 associated with the prehomogenous vector space of symmetric matrices of degree 2.

Classification of irreducible representations of metaplectic covers of the general linear group over a non-archimedean local field

Let F F be a non-archimedean local field and r r a non-negative integer. The classification of the irreducible representations of G L r ( F ) GL_r(F) in terms of supercuspidal representations is one of the highlights of the Bernstein–Zelevinsky theory. We give an analogous classification for metaplectic coverings of G L r ( F ) GL_r(F) .

Classical Theta Lifts for Higher Metaplectic Covering Groups

The classical theta correspondence establishes a relationship between automorphic representations on special orthogonal groups and automorphic representations on symplectic groups or their double covers. This correspondence is achieved by using as integral kernel a theta series on the metaplectic double cover of a symplectic group that is constructed from the Weil representation. There is also an analogous local correspondence. In this work we present an extension of the classical theta correspondence to higher degree metaplectic covers of symplectic and special orthogonal groups. The key issue here is that for higher degree covers there is no analogue of the Weil representation, and additional ingredients are needed. Our work reflects a broader paradigm: constructions in automorphic forms that work for algebraic groups or their double covers should often extend to higher degree metaplectic covers.

Vertex Operators, Solvable Lattice Models and Metaplectic Whittaker Functions

We show that spherical Whittaker functions on an n-fold cover of the general linear group arise naturally from the quantum Fock space representation of $$U_q(\widehat{\mathfrak {sl}}(n))$$ introduced by Kashiwara, Miwa and Stern (KMS). We arrive at this connection by reconsidering the solvable lattice models known as “metaplectic ice” whose partition functions are metaplectic Whittaker functions. First, we show that a certain Hecke action on metaplectic Whittaker coinvariants agrees (up to twisting) with a Hecke action of Ginzburg, Reshetikhin, and Vasserot arising in quantum affine Schur-Weyl duality. This allows us to expand the framework of KMS by Drinfeld twisting to introduce Gauss sums into the quantum wedge, which are necessary for connections to metaplectic forms. Our main theorem interprets the row transfer matrices of this ice model as “half” vertex operators on quantum Fock space that intertwine with the action of $$U_q(\widehat{\mathfrak {sl}}(n))$$ . In the process, we introduce new symmetric functions termed metaplectic symmetric functions and explain how they are related to Whittaker functions on an n-fold metaplectic cover of $${\text {GL}}_r$$ . These resemble LLT polynomials or ribbon symmetric functions introduced by Lascoux, Leclerc and Thibon, and in fact the metaplectic symmetric functions are (up to twisting) specializations of supersymmetric LLT polynomials defined by Lam. Indeed Lam constructed families of symmetric functions from Heisenberg algebra actions on the Fock space commuting with the $$U_q(\widehat{\mathfrak {sl}}(n))$$ -action. The Heisenberg algebra is independent of Drinfeld twisting of the quantum group. We explain that half vertex operators agree with Lam’s construction and this interpretation allows for many new identities for metaplectic symmetric and Whittaker functions, including Cauchy identities. While both metaplectic symmetric functions and LLT polynomials can be related to vertex operators on the quantum Fock space, only metaplectic symmetric functions are connected to solvable lattice models.

On Poincaré series of half-integral weight

We use Poincare series of K-finite matrix coefficients of genuine integrable representations of the metaplectic cover of SL_2(ℝ) to construct a spanning set for the space of cusp forms S_m(Γ, χ), where Γ is a discrete subgroup of finite covolume in the metaplectic cover of SL_2(ℝ), χ is a character of Γ of finite order, and m is in 5/2+ℤ_{; ; ≥0}; ; . We give a result on the non-vanishing of the constructed cusp forms and compute their Petersson inner product with any f in S_m(Γ, χ). Using this last result, we construct a Poincare series Δ_{; ; Γ, k, m, ξ, χ}; ; in S_m(Γ, χ) that corresponds, in the sense of the Riesz representation theorem, to the linear functional f ↦ f^{; ; (k)}; ; (ξ) on S_m(Γ, χ), where ξ is in ℂ_{; ; ℑ(z)>0}; ; and k is in ℤ_{; ; ≥0}; ; . Under some additional conditions on Γ and χ, we provide the Fourier expansion of cusp forms Δ_{; ; Γ, k, m, ξ, χ}; ; and their expansion in a series of classical Poincare series.

Restriction of representations of metaplectic GL2(F) to tori

Let F be a non-archimedean local field. We study the restriction of irreducible admissible genuine representations of the twofold metaplectic cover $${\widetilde {GL}_2}$$ of GL2(F) to the inverse image in $${\widetilde {GL}_2}$$ of a maximal torus in GL2(F).

On the non-vanishing of Poincaré series on the metaplectic group

In this paper, we study the K-finite matrix coefficients of integrable representations of the metaplectic cover of $$ {\mathrm {SL}}_{2}({\mathbb {R}}) $$ and give a result on the non-vanishing of their Poincare series. We do this by adapting the techniques developed for $$ {\mathrm {SL}}_{2}({\mathbb {R}}) $$ by Muic to the case of the metaplectic group.

Branching laws for the metaplectic cover of GL2

Let $F$ be a non-Archimedian local field of characteristic zero and $E/F$ a quadratic extension. The aim of the present article is to study the multiplicity of an irreducible admissible representation of ${\rm GL}_2(F)$ occurring in an irreducible admissible genuine representation of non-trivial two fold covering $\widetilde{{\rm GL}}_2(E)$ of ${\rm GL}_2(E)$.

Irreducible admissible mod-p representations of metaplectic groups

Let p be an odd prime number, and F a nonarchimedean local field of residual characteristic p. We classify the smooth, irreducible, admissible genuine mod-p representations of the twofold metaplectic cover $$\widetilde{\text {Sp}}_{2n}(F)$$ of $$\text {Sp}_{2n}(F)$$ in terms of genuine supercuspidal (equivalently, supersingular) representations of Levi subgroups. To do so, we use results of Henniart–Vigneras as well as new technical results to adapt Herzig’s method to the metaplectic setting. As consequences, we obtain an irreducibility criterion for principal series representations generalizing the complete irreducibility of principal series representations in the rank 1 case, as well as the fact that irreducibility is preserved by parabolic induction from the cover of the Siegel Levi subgroup.

Criteria for the existence of cuspidal theta representations

Theta representations appear globally as the residues of Eisenstein series on covers of groups; their unramified local constituents may be characterized as subquotients of certain principal series. A cuspidal theta representation is one which is equal to the local twisted theta representation at almost all places. Cuspidal theta representations are known to exist but only for covers of $GL_j$, $j\leq 3$. In this paper we establish necessary conditions for the existence of cuspidal theta representations on the $r$-fold metaplectic cover of the general linear group of arbitrary rank.

A converse theorem for metaplectic Eisenstein series on the double and triple covers of SL2(ℚ(−D))

In this work, we prove a converse theorem for metaplectic Eisenstein series on the [Formula: see text]th metaplectic cover of the group [Formula: see text], where [Formula: see text] is an imaginary quadratic number field containing the [Formula: see text]th roots of unity. This is an analog to previous converse theorems relating certain double Dirichlet series to the Mellin transforms of Eisenstein series of half-integer weight. We also propose a way to generalize this result to any number field.

A Question on Splitting of Metaplectic Covers

Let F be a non-Archimedian local field. Splitting of twofold metaplectic cover of Sp2n(F) restricted to various subgroups of Sp2n(F) is important in application of the Weil representation of the metaplectic group. Let E/F be a quadratic extension. In this paper, we prove the splitting of the metaplectic cover of GL2(E) restricted to the subgroup , where DF is the quaternion division algebra with center F, as a first step in our study of the restriction of representations of metaplectic cover of GL2(E) to GL2(F) and . These results were suggested to the author by Professor Dipendra Prasad.

Metaplectic Tensor Products for Automorphic Representation of (r)

AbstractLet M = GLr1 ✗ … × GLrk ⊆ GLr be a Levi subgroup of GLr, where r = r1 + … +rk, and its metaplectic preimage in the n-fold metaplectic cover r1 of GLr. For automorphic representations π1, …, πk of r1 (), … ,rk (), we construct (under a certain technical assumption that is always satisfied when n = 2) an automorphic representation π of () that can be considered as the “tensor product” of the representations π1, … , πk. This is the global analogue of the metaplectic tensor product defined by P. Mezo in the sense that locally at each place v, πv is equivalent to the local metaplectic tensor product of π1,v, … , πk,v defined by Mezo. Then we show that if all of the πi are cuspidal (resp. square-integrable modulo center), then the metaplectic tensor product is cuspidal (resp. square-integrable modulo center). We also show that (both locally and globally) the metaplectic tensor product behaves in the expected way under the action of a Weyl group element and show the compatibility with parabolic inductions.

Short incomplete Gauss sums and rational points on metaplectic horocycles

In this paper, we investigate the limiting behavior of short incomplete Gauss sums at random argument as the number of terms goes to infinity. We prove that the limit distribution is given by the distribution of theta sums and differs from the limit law for long Gauss sums studied by the author and Marklof. The key ingredient in the proof is an equidistribution theorem for rational points on horocycles in the metaplectic cover of SL(2, ℝ).

Metaplectic Demazure operators and Whittaker functions

In [CG10] the first two named authors defined an action of a Weyl group on rational functions and used it to construct multiple Dirichlet series. These series are related to Whittaker functions on an n-fold metaplectic cover of a reductive group. In this paper, we define metaplectic analogues of the Demazure and Demazure-Lusztig operators. We show how these operators can be used to recover the formulas from [CG10], and how, together with results of McNamara [McN], they can be used to compute Whittaker functions on metaplectic groups over p-adic fields.

A metaplectic Casselman-Shalika formula for GL r

We provide formulas for various bases of spherical Whittaker functions on the $n$-fold metaplectic cover of ${\rm GL}_r$ over a $p$-adic field and show that there is a basis of symmetric functions in the complex parameter. In addition we relate a specific spherical Whittaker function to the $p$-power part of the Weyl group multiple Dirichlet series for the root system of type $A_{r-1}$ constructed from $n$th order Gauss sums. We also show that the zonal spherical functions can be computed explicitly in terms of Hall-Littlewood polynomials as in Macdonald's formula for ${\rm GL}_r$.

Weil representations associated with finite quadratic modules

To any finite quadratic module, that is, a finite abelian group together with a non-degenerate quadratic form, it is possible to associate a representation of $$\mathrm{Mp}_{2}(\mathbb Z )$$ , the metaplectic cover of the modular group. This representation is usually referred to as a Weil representation and our main result is a general explicit formula for its matrix coefficients. This result completes earlier work by Scheithauer in the case when the representation factors through $$\mathrm{SL}_{2}(\mathbb Z )$$ . Furthermore, our formula is given in a such a way that it is easy to implement efficiently on a computer.

A Bessel identity for the theta correspondence

We establish a spectral identity between global Bessel distributions with respect to generic cuspidal representations of an odd orthogonal group and the metaplectic cover of a symplectic group which are related by the theta correspondence. We also provide analogous local identities for square-integrable representations.

Weyl group multiple Dirichlet series, Eisenstein series and crystal bases

We show that the Whittaker coecients of Borel Eisenstein series on the metaplectic covers of GLr+1 can be described as multiple Dirichlet series in r complex variables, whose coecients are computed by attaching a number-theoretic quantity (a product of Gauss sums) to each vertex in a crystal graph. These Gauss sums depend on \string previously introduced in work of Lusztig, Berenstein and Zelevinsky, and Littelmann. These data are the lengths of segments in a path from the given vertex to the vertex of lowest weight, depending on a factorization of the long Weyl group element into simple reections. The coecients may also be described as sums over strict Gelfand-Tsetlin patterns. The description is uniform in the degree of the metaplectic cover.

EXPLICIT DOUBLING INTEGRALS FOR Sp2(F) AND $\widetilde{{{\rm Sp}}_{2}}(F)$ USING "GOOD TEST VECTORS"

In this paper, we offer some explicit computations of a formulation of the doubling method of Piatetski-Shapiro and Rallis for the groups Sp 2(F) (the rank 2 symplectic group) and its metaplectic cover [Formula: see text] for F a finite extension of ℚp with p ≠ 2. We determine a set of "good test vectors" for the irreducible constituents of unramified principal series representations for these groups as well as a set of "good theta test sections" in a family of degenerate principal series representations of Sp 4(F) and [Formula: see text]. Determining "good test data" that produces a non-vanishing doubling integral should indicate the existence of a non-vanishing theta lifts for dual pairs of the type ( Sp 2(F), O (V)) (respectively [Formula: see text]) where V is a quadratic space of an even (respectively odd) dimension.