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Gordan Savin

Publications and source records attributed to Gordan Savin.

At least 19 recordsLinked to original sources
arXiv2026

Long-root $\SL_2$ dual pairs in split exceptional groups: Compact types, cone geometry, and theta correspondence

The paper establishes a case of Howe duality for an exceptional theta correspondence using the ping-pong of types method. To that end, one needs to compute lifts of types of maximal compact subgroups. A brand new ingredient enters here: The graded module of the minimal representations is the coordinate ring of an affine cone whose projectivization is a Grassmannian. Via the filtered Nakayama lemma the problem of lifting types becomes a problem in algebraic geometry: compute certain linear sections of the Grassmannian.

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arXiv2026

$\SL_3$ dual pairs

We study the exceptional theta correspondence for dual pairs of the form $\SL_3\times G$ inside $p$-adic groups of type $E_n$, $n=6,7,8$. We establish Howe duality for lifts of all Whittaker-generic unitarizable representations of $\SL_3$. Our dual pair is in a see-saw relationship with a dual pair of the form $G_2 \times H$ such that $G/H$ is a rank-two symmetric space. Using the theta correspondence for $\SL_3 \times G$, we prove a multiplicity-one result for $H$-invariant functionals on representations of $G$.

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arXiv2026

Exceptional dual pair correspondences; case of real groups of split rank one

Exceptional real groups have quaternionic forms of split rank 4 that contain dual pairs $G\times G'$, where $G'$ is the split Lie group of the type $G_2$, and $G$ a Lie group of split rank one. In this paper we restrict the minimal representation of the quaternionic group to the dual pair and prove some significant results for the resulting correspondence of representations.

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arXiv2025

On the classification of unitary highest weight modules in the exceptional cases

In our previous paper, we gave a complete classification of the unitary highest weight modules for the universal covers of the Lie groups $Sp(2n, \mathbb{R}), SO^{*}(2n)$ and $SU(p, q)$, using the Dirac inequality and the so called PRV product. In this paper, we complete the classification of the unitary highest weight modules for the remaining cases; i.e., universal covers of the Lie groups $SO_{e}(2, n)$, $E_{6(-14)}$ and $E_{7(-25)}$. We also describe unitary highest weight modules with given infinitesimal characters.

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arXiv2023

The Dual Pair $\mathrm{Aut}(C)\times F_{4}$ ($p$-adic case)

We study the local theta correspondence for dual pairs of the form $\mathrm{Aut}(C)\times F_{4}$ over a $p$-adic field, where $C$ is a composition algebra of dimension 2 or 4, by restricting the minimal representation of a group of type $E$. We investigate this restriction through the computation of maximal parabolic Jacquet modules and the Fourier-Jacobi functor. As a consequence of our results we prove a multiplicity one result for the $\mathrm{Spin}(9)$-invariant linear functionals of irreducible representations of $F_{4}$ and classify the $\mathrm{Spin}(9)$-distinguished representations.

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arXiv2023

Similitude exceptional theta correspondences

We construct and develop a similitude version of exceptional theta correspondences and show that the Howe duality theorem follows from that for the "isometry" case. We also extend basic tools such as the seesaw identity associated to seesaw dual pairs to the similitude setting.

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arXiv2023

A theory of $γ$-factors for $G_2 \times GL_r$

We construct a theory of local gamma factors for $G_2 \times GL_r$ using a functorial lifting from $G_2$ to $GL_7$. This theory of gamma factors is uniquely characterized by a usual list of properties, showing that it is the only possible candidate. Moreover, this theory of gamma factors is compatible with the Galois theoretic one under the local Langlands correspondence for $G_2$.

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arXiv2023

A family of Spin(8) dual pairs: the case of real groups

Exceptional groups of type $E_6$ contain dual pairs where one member is $\mathrm{Spin}(8)$, and the other is $T\rtimes \mathbb Z/2\mathbb Z$, where $T$ is a two-dimensional torus and the non-trivial element in $\mathbb Z/2\mathbb Z$ acts on $T$ by the inverse involution. We describe the correspondence of representations arising by restricting the minimal representation.

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arXiv2021

Howe duality and dichotomy for exceptional theta correspondences

We study three exceptional theta correspondences for p-adic groups, where one member of the dual pair is the exceptional group G2. We prove the Howe duality conjecture for these dual pairs and a dichotomy theorem, and determine explicitly the theta lifts of all non-cuspidal representations.

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arXiv2020

The Gelfand--Graev representation of SO$(2n+1)$ in terms of Hecke algebras

Let $G$ be a $p$-adic classical group. The representations in a given Bernstein component can be viewed as modules for the corresponding Hecke algebra---the endomorphism algebra of a pro-generator of the given component. Using Heiermann's construction of these algebras, we describe the Bernstein components of the Gelfand--Graev representation for $G=$SO$(2n+1)$.

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arXiv2020

Euler-Poincaré formulae for positive depth Bernstein projectors

Work of Bezrukavnikov-Kazhdan-Varshavsky uses an equivariant system of trivial idempotents of Moy-Prasad groups to obtain an Euler-Poincaré formula for the r-depth Bernstein projector. Barbasch-Ciubotaru-Moy use depth-zero cuspidal representations of parahoric subgroups to decompose the Euler-Poincaré presentation of the depth-zero projector. For positive depth $r$, we establish a decomposition of the Euler-Poincaré presentation of the r-depth Bernstein projector based on a notion of associate classes of cuspidal pairs for Moy-Prasad quotients. We apply these new Euler-Poincaré presentations to the obtain decompositions of the resolutions of Schneider-Stuhler and Bestvina-Savin.

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