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Yuval Z. Flicker

Publications and source records attributed to Yuval Z. Flicker.

11 recordsLinked to original sources
arXiv2018

Affine quantum super Schur-Weyl duality

The Schur-Weyl duality, which started as the study of the commuting actions of the symmetric group $S_d$ and $\mathrm{GL}(n,\mathbb{C})$ on $V^{\otimes d}$ where $V=\mathbb{C}^n$, was extended by Drinfeld and Jimbo to the context of the finite Iwahori-Hecke algebra $H_d(q^2)$ and quantum algebras $U_q(\mathrm{gl}(n))$, on using universal $R$-matrices, which solve the Yang-Baxter equation. There were two extensions of this duality in the Hecke-quantum case: to the affine case, by Chari and Pressley, and to the super case, by Moon and by Mitsuhashi. We complete this chain of works by completing the cube, dealing with the general affine super case, relating the commuting actions of the affine Iwahori-Hecke algebra $H^a_d(q^2)$ and of the affine quantum Lie superalgebra $U_{q,a}^σ(\mathrm{sl}(m,n))$ using the presentation by Yamane in terms of generators and relations, acting on the $d$th tensor power of the superspace $V=\mathbb{C}^{m+n}$. Thus we construct a functor and show it is an equivalence of categories of $H_d^a(q^2)$ and $U_{q,a}^σ(\mathrm{sl}(m,n))$-modules when $d<n'=m+n$.

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arXiv2008

On Level-Raising Congruences

A work of Sorensen is rewritten here to include nontrivial types at the infinite places. This extends results of K. Ribet and R. Taylor on level-raising for algebraic modular forms on D^{\times}, where D is a definite quaternion algebra over a totally real field F. This is done for any automorphic representations πof an arbitrary reductive group G over F which is compact at infinity. It is not assumed that π_\infty is trivial. If λis a finite place of \bar{\Q}, and w is a place where π_w is unramified and π_w is congruent to the trivial representation mod λ, then under some mild additional assumptions (relaxing requirements on the relation between w and \ell which appear in previous works) the existence of a \tildeπ congruent to πmod λsuch that \tildeπ_w has more parahoric fixed vectors than π_w, is proven. In the case where G_w has semisimple rank one, results of Clozel, Bellaiche and Graftieaux according to which \tildeπ_w is Steinberg, are sharpened. To provide applications of the main theorem two examples over F of rank greater than one are considered. In the first example G is taken to be a unitary group in three variables and a split place w. In the second G is taken to be an inner form of GSp(2). In both cases, precise satisfiable conditions on a split prime w guaranteeing the existence of a \tildeπ congruent to πmod λsuch that the component \tildeπ_w is generic and Iwahori spherical, are obtained. For symplectic G, to conclude that \tildeπ_w is generic, computations of R. Schmidt are used. In particular, if πis of Saito-Kurokawa type, it is congruent to a \tildeπ which is not of Saito-Kurokawa type.

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arXiv2008

Cyclic odd degree base change lifting for unitary groups in three variables

Let E/F be a quadratic number (resp. p-adic) field extension, and F' an odd degree cyclic field extension of F. We establish a base-change functorial lifting of automorphic (resp. admissible) representations from the unitary group U(3,E/F) associated with E/F to the unitary group U(3,F'E/F'). As a consequence, we classify the invariant packets of U(3,F'E/F'), namely those which contain (irreducible) automorphic (resp. admissible) representations which are invariant under the action of the Galois group Gal(F'E/E). To do this we use the trace formula technique, and well-known results on the base-change lifting from U(3,E/F) to GL(3,E) and on the base-change lifting for the general linear groups. We also determine the invariance of individual representations, using Howe correspondence. This work is the first study of base change into an algebraic group whose packets are not all singletons, and which does not satisfy the "strong multiplicity one theorem." Novel phenomena are encountered: e.g. there are invariant packets where not every irreducible automorphic (resp. admissible) member is Galois-invariant. We also obtain local twisted character identities with respect to this base-change lifting.

math.NT↗
arXiv2007

On the symmetric square. Unstable Twisted Characters

We provide a purely local computation of the (elliptic) twisted (by "transpose-inverse") character of the representation π=I(\1) of PGL(3) over a p-adic field induced from the trivial representation of the maximal parabolic subgroup. This computation is independent of the theory of the symmetric square lifting of [IV] of automorphic and admissible representations of SL(2) to PGL(3). It leads to a proof of the (unstable) fundamental lemma in the theory of the symmetric square lifting, namely that corresponding spherical functions (on PGL(2) and PGL(3)) are matching: they have matching orbital integrals.

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arXiv2007

On poles of twisted tensor L-functions

It is shown that the only possible pole of the twisted tensor L-functions in Re(s)\geq 1 is located at s=1 for all quadratic extensions of global fields.

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arXiv2006

Twisted character of a small representation of PGL(4)

We compute by a purely local method the elliptic, twisted by transpose-inverse, character χ_πof the representation π=I_{(3,1)}(1_3) of PGL(4,F) normalizedly induced from the trivial representation of the maximal parabolic subgroup of type (3,1), where F is a p-adic field. Put C=(GL(2,F)xGL(2,F))'/F^x (F^x embeds diagonally, prime means equal determinants). It is a twisted elliptic endoscopic group of PGL(4). We deduce from the computation that χ_πis an unstable function: its value at one twisted regular elliptic conjugacy class with norm in C is minus its value at the other class within the twisted stable conjugacy class, and zero at the classes without norm in C. Moreover πis the unstable endoscopic lift of the trivial representation of C. Naturally, this computation plays a role in the theory of lifting from C (=``SO(4,F)'') and PGp(2,F) to PGL(4,F) using the trace formula. Our work develops a 4-dimensional analogue of the model of the small representation of PGL(3,F) introduced by the first author with Kazhdan in a 3-dimensional case, and it uses the classification of twisted stable and unstable regular conjugacy classes in PGL(4,F). It extends the local method of computation introduced by us in the 3-dimensional case. An extension math.NT/0606263 of our work here to apply to similar representations of GL(4,F) whose central character is nontrivial will appear in Int. J. Number Theory.

math.NT↗
arXiv2006

Twisted character of a small representation of GL(4)

We compute by a purely local method the (elliptic) twisted by transpose-inverse character χ_{π_Y} of the representation π_Y=I_{(3,1)}(1_3xχ_Y) of G=GL(4,F), where F is a p-adic field, p not 2, and Y is an unramified quadratic extension of F; χ_Y is the nontrivial character of F^\x/N_{Y/F}Y^x. The representation π_Y is normalizedly induced from \pmatrix m_3&\ast 0&m_1\endpmatrix \mapstoχ_Y(m_1), m_i in GL(i,F), on the maximal parabolic subgroup of type (3,1). We show that the twisted character χ_{π_Y} of π_Y is an unstable function: its value at a twisted regular elliptic conjugacy class with norm in C_Y=``GL(2,Y)/F^x'' is minus its value at the other class within the twisted stable conjugacy class. It is zero at the classes without norm in C_Y. Moreover π_Y is the endoscopic lift of the trivial representation of C_Y. We deal only with unramified Y/F, as globally this case occurs almost everywhere. Naturally this computation plays a role in the theory of lifting of C_Y and GSp(2) to GL(4) using the trace formula. Our work extends -- to the context of nontrivial central characters -- the work of math.NT/0606262, where representations of PGL(4,F) are studied. In math.NT/0606262 a 4-dimensional analogue of the model of the small representation of PGL(3,F) introduced with Kazhdan in a 3-dimensional case is developed, and the local method of computation introduced by us in the 3-dimensional case is extended. As in math.NT/0606262 we use here the classification of twisted (stable) regular conjugacy classes in GL(4,F).

math.NT↗
arXiv2006

Characters, genericity, and multiplicity one for U(3)

Publications on automorphic representations of the group U(3) assumed the validity of multiplicity one theorem since I claimed it in 1982. But the argument, published 1988, was based on a misinterpretation of a claim of Gelbart and Piatetski-Shapiro, SLN 1041 (1984), Prop. 2.4(i): ``L^2_{0,1} has multiplicity 1'', as meaning that each irreducible in the space L^2_{0,1} of generic cusp forms has multiplicity one in the space L^2_0 of cusp forms. The statement meant in SLN 1041, ``each irreducible in L^2_{0,1} has multiplicity 1 in L^2_{0,1}'' is too weak to be useful for nongeneric representations, as the present article points out. To remedy the situation, we detail in this paper the local method we sketched in the paper of 1988. Let ψbe a generic character of the unipotent radical U of a Borel subgroup of a quasisplit p-adic group G. The number (0 or 1) of ψ-Whittaker models on an admissible irreducible representation πof G was expressed by Rodier in terms of the limit of values of the trace of πat certain measures concentrated near the origin. An analogous statement holds in the twisted case. This article proves this twisted analogue for an involution when G=U(3) and p is not 2, and uses it to provide a local proof of the multiplicity one theorem for U(3). This asserts that each discrete spectrum automorphic representation (with induced components at the dyadic places) of the quasisplit unitary group U(3) associated with a quadratic extension E/F of number fields occurs in the discrete spectrum with multiplicity one.

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arXiv2000

Motivic torsors

The torsor P_s=Hom(H_{\DR},H_s) under the motivic Galois group G_s=Aut H_s of the Tannakian category M_k generated by one-motives related by absolute Hodge cycles over a field k with an embedding s into the complex numbers is shown to be determined by its global projection [P_s\to (P_s)/(G_s)^0] to a Gal(\ov k/k)-torsor, and by its localizations (P_s) x_k (k_ξ) at a dense subset of orderings ξof the field k, provided k has virtual cohomological dimension (vcd) one. This result is an application of a recent local-global principle for connected linear algebraic groups over a field k of vcd=1.

math.AG↗
arXiv1998

Grothendieck's theorem on non-abelian H^2 and local-global principles

A theorem of Grothendieck asserts that over a perfect field k of cohomological dimension one, all non-abelian H^2-cohomology sets of algebraic groups are trivial. The purpose of this paper is to establish a formally real generalization of this theorem. The generalization -- to the context of perfect fields of virtual cohomological dimension one -- takes the form of a local-global principle for the H^2-sets with respect to the orderings of the field. This principle asserts in particular that an element in H^2 is neutral precisely when it is neutral in the real closure with respect to every ordering in a dense subset of the real spectrum of k. Our techniques provide a new proof of Grothendieck's original theorem. An application to homogeneous spaces over k is also given.

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