results for au:Savin_G in:math

- Aug 15 2017 math.RT arXiv:1708.03707v1We consider affine buildings with refined chamber structure. For each vertex in the refined chamber structure we construct a contraction, based at the vertex, that is used to prove exactness of Schneider-Stuhler resolutions of arbitrary depth.
- Jul 31 2017 math.NT arXiv:1707.09001v1Fix a quadratic order over the ring of integers. An embedding of the quadratic order into a quaternionic order naturally gives an integral binary hermitian form over the quadratic order. We show that, in certain cases, this correspondence is a discriminant preserving bijection between the isomorphism classes of embeddings and integral binary hermitian forms.
- We study the exceptional theta correspondence for real groups obtained by restricting the minimal representation of the split exceptional group of the type E_n, to a split dual pair where one member is the exceptional group of the type G_2. We prove that the correspondence gives a bijection between spherical representations if n=6,7, and a slightly weaker statement if n=8.
- Let $G$ be a general linear group over a $p$-adic field. It is well known that Bernstein components of the category of smooth representations of $G$ are described by Hecke algebras arising from Bushnell-Kutzko types. We describe the Bernstein components of the Gelfand-Graev representation of $G$ by explicit Hecke algebra modules. This result is used to translate the theory of Bernstein-Zelevinsky derivatives in the language of representations of Hecke algebras, where we develop a comprehensive theory.
- Let $G$ be a split reductive group over a $p$-adic field $F$. Let $B$ be a Borel subgroup and $U$ the maximal unipotent subgroup of $B$. Let $\psi$ be a Whittaker character of $U$. Let $I$ be an Iwahori subgroup of $G$. We describe the Iwahori-Hecke algebra action on the Gelfand-Graev representation $(\mathrm{ind}_{U}^{G}\psi)^I$ by an explicit projective module. As a consequence, for $G=GL(n,F)$, we define and describe Bernstein-Zelevinsky derivatives of representations generated by $I$-fixed vectors in terms of the corresponding Iwahori-Hecke algebra modules. Furthermore, using Lusztig's reductions, we show that the Bernstein-Zelevinsky derivatives can be determined using graded Hecke algebras. We give two applications of our study. Firstly, we compute the Bernstein-Zelevinsky derivatives of generalized Speh modules, which recovers a result of Lapid-Mínguez and Tadić. Secondly, we give a realization of the Iwahori-Hecke algebra action on some generic representations of $GL(n+1,F)$, restricted to $GL(n,F)$, which is further used to verify a conjecture on an Ext-branching problem of D. Prasad for a class of examples.
- Apr 27 2016 math.RT arXiv:1604.07794v2Suppose $(G,G')$ is a dual pair of subgroups of a metaplectic group. The dual pair correspondence is a bijection between (subsets of the) irreducible representations of $G$ and $G'$, defined by the non-vanishing of Hom$(\omega,\pi\times\pi')$, where $\omega$ is the oscillator representation. Alternatively one considers Hom$_G(\omega,\pi)$ as a $G'$-module. It is fruitful to replace Hom with Ext$^i$, and general considerations suggest that the Euler-Poincare characteristic EP$(\omega,\pi)$, the alternating sum of Ext$^i(\omega,\pi)$, will be a more elementary object. We restrict to the case of $p$-adic groups, and prove that EP$(\omega,\pi)$ is a well defined element of the Grothendieck group of finite length representations of $G'$, and show that it is indeed more elementary than Hom$(\omega,\pi)$. We expect that computation of EP, together with vanishing results for higher Ext groups, will be a useful tool in computing the dual pair correspondence, and will help to elucidate the structure of Hom$(\omega,\pi)$.
- Nov 17 2015 math.RT arXiv:1511.04821v1For maximal compact subgroups of the metaplectic group, the minimal types in the Schrödinger model of the Weil representation are calculated explicitly. Although these types are known in the case of odd residual characteristic, this computation is done for arbitrary residual characteristic.
- Jan 29 2015 math.RT arXiv:1501.07069v2In this paper we study the local theta correspondences between epipelagic supercupsidal representations of a type I classical dual pair $(G,G')$ over $p$-adic fields. We show that, besides an exceptional case, an epipelagic supercupsidal representation $\pi$ of $\widetilde{G}$ lifts to an epipelagic supercupsidal representation $\pi'$ of $\widetilde{G}'$ if and only if the epipelagic data of $\pi$ and $\pi'$ are related by the moment maps.
- We study wave-front sets of representations of reductive groups over global or non-archimedean local fields.
- We prove that automorphic representations whose local components are certain small representations have multiplicity one. The proof is based on the multiplicity-one theorem for certain functionals of small representations, also proved in this paper.
- Jun 17 2014 math.NT arXiv:1406.3773v1We construct extensions of the field of rational numbers with the Galois group G_2(F_p) by reducing p-adic representations attached to automorphic representations.
- Let G be a reductive group and P=MN a maximal parabolic subgroup. The group M acts, by conjugation, on N/[N,N]. It is well known that, over an algebraically closed field, the group M acts transitively on a Zariski open set. However, over a general field, the structure of orbits may be quite non-trivial. A description may involve unexpected invariants. A notable example is when G is a split, simply connected group of type D_4, and P is the maximal parabolic corresponding to the branching point of the Dynkin diagram. The space N/[N,N] is also known as the Bhargava cube, and it was the starting point of his investigations of prehomogeneous spaces. We consider a version of this problem for the triality D_4.
- Let $\mathbf{G}$ be a split algebrac group of type $E_n$ defined over a $p$-adic field. This group contains a dual pair $G \times G'$ where one of the groups is of type $G_2$. The minimal representation of $\mathbf{G}$, when restricted to the dual pair, gives a correspondence of representations of the two groups in the dual pair. We prove a matching of spherical Hecke algebras of $G$ and $G'$, when acting on the minimal representation. This implies that the correspondence is functorial, in the sense of Arthur and Langlands, for spherical representations.
- We show that every exceptional Lie algebra over a number field can be obtained by Tits' construction from an octonion algebra O and a cubic Jordan algebra J. In particular, the exceptional Lie algebra contains a dual pair which is the direct sum of the derivation algebras of O and J. We determine rational forms of this dual pair.
- We use Bernstein's presentation of the Iwahori-Matsumoto Hecke algebra to obtain a simple proof of the Satake isomorphism and, in the same stroke, compute the center of the Iwahori-Matsumoto Hecke algebra.
- We generalize Conway's approach to integral binary quadratic forms on Q to study integral binary hermitian forms on quadratic imaginary extensions of Q. In Conway's case, an indefinite form that doesn't represent 0 determines a line ("river") in the spine T associated with SL(2,Z) in the hyperbolic plane. In our generalization, such a form determines a plane ("ocean") in Mendoza's spine associated with the corresponding Bianchi group SL(2,A) in hyperbolic 3-space.
- The local Langlands conjectures imply that to every generic supercuspidal irreducible representation of $G_2$ over a $p$-adic field, one can associate a generic supercuspidal irreducible representation of either $PGSp_6$ or$PGL_3$. We prove this conjectural dichotomy, demonstrating a precise correspondence between certain representations of $G_2$ and other representations of $PGSp_6$ and $PGL_3$. This correspondence arises from theta correspondences in $E_6$ and $E_7$, analysis of Shalika functionals, and spin L-functions. Our main result reduces the conjectural Langlands parameterization of generic supercuspidal irreducible representations of $G_2$ to a single conjecture about the parameterization for $PGSp_6$.
- In this paper we develop representation theory of the two fold central extension of SL(2) over the field of 2-adic numbers. As a consequence, we obtain a local Shimura correspondence between the two fold central extension of SL(2) and the linear group PGL(2), which we show to be compatible with the global Shimura correspondence at p=2.
- Jul 08 2008 math.NT arXiv:0807.0861v1For every finite field F and every positive integer r, there exists a finite extension F' of F such that either SO(2r+1,F') or its simple derived group can be realized as a Galois group over Q. If the characteristic of F is 3 or 5 (mod 8), then we can guarantee that the derived group of SO(2r+1,F') can be realized. Likewise, for every finite field F, there exists a finite extension F' of F such that the finite simple group G_2(F') can be realized a Galois group over Q. The proof uses automorphic forms to construct Galois representations which cut out Galois extensions of the desired type.
- Oct 30 2006 math.NT arXiv:math/0610860v3We prove that there are infinitely many finite simple groups of symplectic Lie type, of any specified characteristic and rank, which appear as Galois groups over the field of rational numbers. This generalizes a result of Wiese, which inspired this paper.
- May 24 1995 math.RT arXiv:math/9505209v1Let $G$ be a split simply laced group defined over a $p$-adic field $F$. In this paper we study the restriction of the minimal representation of $G$ to various dual pairs in $G$. For example, the restriction of the minimal representation of $E_7$ to the dual pair $G_2 \times{}$Sp(6) gives the non-endoscopic Langlands lift of irreducible representations of $G_2$ to Sp(6).