arXiv subjects

Siddhartha Sahi

Publications and source records attributed to Siddhartha Sahi.

At least 37 records · Page 2Linked to original sources

arXiv2023

Askey-Wilson Polynomials and Branching Laws

Connection coefficient formulas for special functions describe change of basis matrices under a parameter change, for bases formed by the special functions. Such formulas are related to branching questions in representation theory. The Askey-Wilson polynomials are one of the most general 1-variable special functions. Our main results are connection coefficient formulas for shifting one of the parameters of the nonsymmetric Askey-Wilson polynomials. We also show how one of these results can be used to re-prove an old result of Askey and Wilson in the symmetric case. The method of proof combines establishing a simpler special case of shifting one parameter by a factor of q with using a co-cycle condition property of the transition matrices involved. Supporting computations use the Noumi representation and are based on simple formulas for how some basic Hecke algebra elements act on natural almost symmetric Laurent polynomials.

math.CA↗

arXiv2023

From Cauchy's determinant formula to bosonic and fermionic immanant identities

Cauchy's determinant formula (1841) involving $\det ((1-u_i v_j)^{-1})$ is a fundamental result in symmetric function theory. It has been extended in several directions, including a determinantal extension by Frobenius [J. reine angew. Math. 1882] involving a sum of two geometric series in $u_i v_j$. This theme also resurfaced in a matrix analysis setting in a paper by Horn [Trans. Amer. Math. Soc. 1969] - where the computations are attributed to Loewner - and in recent works by Belton-Guillot-Khare-Putinar [Adv. Math. 2016] and Khare-Tao [Amer. J. Math. 2021]. These formulas were recently unified and extended in [Trans. Amer. Math. Soc. 2022] to arbitrary power series, with commuting/bosonic variables $u_i, v_j$. In this note we formulate analogous permanent identities, and in fact, explain how all of these results are a special case of a more general identity, for any character - in fact, any complex class function - of any finite group that acts on the bosonic variables $u_i$ and on the $v_j$ via signed permutations. (We explain why larger linear groups do not work, via a - perhaps novel - "symmetric function" characterization of signed permutation matrices that holds over any integral domain.) We then provide fermionic analogues of these formulas, as well as of the closely related Cauchy product identities.

math.RT↗

arXiv2022

Weyl algebras for quantum homogeneous spaces

We present a new family of quantum Weyl algebras where the polynomial part is the quantum analog of functions on homogeneous spaces corresponding to symmetric matrices, skew symmetric matrices, and the entire space of matrices of a given size. The construction uses twisted tensor products and their deformations combined with invariance properties derived from quantum symmetric pairs. These quantum Weyl algebras admit $U_q(\mathfrak{gl}_N)$-module algebra structures compatible with standard ones on the polynomial part, have relations that are expressed nicely via matrices, and are closely related to an algebra arising in the theory of quantum bounded symmetric domains.

math.QA↗

arXiv2022

Random and mean Lyapunov exponents for $\mathrm{GL}_n(\mathbb{R})$

We consider orthogonally invariant probability measures on $\mathrm{GL}_n(\mathbb{R})$ and compare the mean of the logs of the moduli of eigenvalues of the matrices to the Lyapunov exponents of random matrix products independently drawn with respect to the measure. We give a lower bound for the former in terms of the latter. The results are motivated by Dedieu-Shub\cite{DS}. A novel feature of our treatment is the use of the theory of spherical polynomials in the proof of our main result.

math.DS↗

arXiv2022

On the extension of the FKG inequality to $n$ functions

The 1971 Fortuin-Kasteleyn-Ginibre (FKG) inequality for two monotone functions on a distributive lattice is well known and has seen many applications in statistical mechanics and other fields of mathematics. In 2008 one of us (Sahi) conjectured an extended version of this inequality for all $n>2$ monotone functions on a distributive lattice. Here we prove the conjecture for two special cases: for monotone functions on the unit square in ${\mathbb R}^k$ whose upper level sets are $k$-dimensional rectangles, and, more significantly, for arbitrary monotone functions on the unit square in ${\mathbb R}^2$. The general case for ${\mathbb R}^k, k>2$ remains open.

math.PR↗

arXiv2021

Interpolation polynomials, bar monomials, and their positivity

We prove a positivity result for interpolation polynomials that was conjectured by Knop and Sahi. These polynomials were first introduced by Sahi in the context of the Capelli eigenvalue problem for Jordan algebras, and were later shown to be related to Jack polynomials by Knop-Sahi and Okounkov-Olshanski. The positivity result proved here is an inhomogeneous generalization of Macdonald's positivity conjecture for Jack polynomials. We also formulate and prove the non-symmetric version of the Knop-Sahi conjecture, and in fact we deduce everything from an even stronger positivity result. This last result concerns certain inhomogeneous analogues of ordinary monomials that we call bar monomials. Their positivity involves in an essential way a new partial order on compositions that we call the bar order, and a new operation that we call a glissade.

math.CO↗

arXiv2021

A reduction principle for Fourier coefficients of automorphic forms

We consider a general class of Fourier coefficients for an automorphic form on a finite cover of a reductive adelic group ${\bf G}(\mathbb{A}_{\mathbb{K}})$, associated to the data of a `Whittaker pair'. We describe a quasi-order on Fourier coefficients, and an algorithm that gives an explicit formula for any coefficient in terms of integrals and sums involving higher coefficients. The maximal elements for the quasi-order are `Levi-distinguished' Fourier coefficients, which correspond to taking the constant term along the unipotent radical of a parabolic subgroup, and then further taking a Fourier coefficient with respect to a $\mathbb{K}$-distinguished nilpotent orbit in the Levi quotient. Thus one can express any Fourier coefficient, including the form itself, in terms of higher Levi-distinguished coefficients. In follow-up papers we use this result to determine explicit Fourier expansions of minimal and next-to-minimal automorphic forms on split simply-laced reductive groups, and to obtain Euler product decompositions of their top Fourier coefficients.

math.NT↗

arXiv2021

Eulerianity of Fourier coefficients of automorphic forms

We study the question of Eulerianity (factorizability) for Fourier coefficients of automorphic forms, and we prove a general transfer theorem that allows one to deduce the Eulerianity of certain coefficients from that of another coefficient. We also establish a `hidden' invariance property of Fourier coefficients. We apply these results to minimal and next-to-minimal automorphic representations, and deduce Eulerianity for a large class of Fourier and Fourier-Jacobi coefficients. In particular, we prove Eulerianity for parabolic Fourier coefficients with characters of maximal rank for a class of Eisenstein series in minimal and next-to-minimal representations of groups of ADE-type that are of interest in string theory.

math.NT↗

arXiv2021

Capelli operators for spherical superharmonics and the Dougall-Ramanujan identity

Let $(V,ω)$ be an orthosympectic $\mathbb Z_2$-graded vector space and let $\mathfrak g:=\mathfrak{gosp}(V,ω)$ denote the Lie superalgebra of similitudes of $(V,ω)$. When the space $\mathscr P(V)$ of superpolynomials on $V$ is \emph{not} a completely reducible $\mathfrak g$-module, we construct a natural basis $D_λ$ of Capelli operators for the algebra of $\mathfrak g$-invariant superpolynomial superdifferential operators on $V$, where the index set $\mathcal P$ is the set of integer partitions of length at most two. We compute the action of the operators $D_λ$ on maximal indecomposable components of $\mathscr P(V)$ explicitly, in terms of Knop-Sahi interpolation polynomials. Our results show that, unlike the cases where $\mathscr P(V)$ is completely reducible, the eigenvalues of a subfamily of the $D_λ$ are \emph{not} given by specializing the Knop-Sahi polynomials. Rather, the formulas for these eigenvalues involve suitably regularized forms of these polynomials. In addition, we demonstrate a close relationship between our eigenvalue formulas for this subfamily of Capelli operators and the Dougall-Ramanujan hypergeometric identity. We also transcend our results on the eigenvalues of Capelli operators to the Deligne category $\mathsf{Rep}(O_t)$. More precisely, we define categorical Capelli operators $\{\mathbf D_{t,λ}\}_{λ\in\mathcal P}^{}$ that induce morphisms of indecomposable components of symmetric powers of $\mathsf V_t$, where $\mathsf V_t$ is the generating object of $\mathsf{Rep}(O_t)$. We obtain formulas for the eigenvalue polynomials associated to the $\left\{\mathbf D_{t,λ}\right\}_{λ\in\mathcal P}$ that are analogous to our results for the operators $\{D_λ\}_{λ\in\mathcal P}^{}$.

math.RT↗

arXiv2020

Hopf-Hecke algebras, infinitesimal Cherednik algebras, and Dirac cohomology

Hopf-Hecke algebras and Barbasch-Sahi algebras were defined by the first named author (2016) in order to provide a general framework for the study of Dirac cohomology. The aim of this paper is to explore new examples of these definitions and to contribute to their classification. Hopf-Hecke algebras are distinguished by an orthogonality condition and a PBW property. The PBW property for algebras such as the ones considered here has been of great interest in the literature and we extend this discussion by further results on the classification of such deformations and by a class of hitherto unexplored examples. We study infinitesimal Cherednik algebras of $GL_n$ as defined by Etingof, Gan, and Ginzburg in [Transform. Groups, 2005] as new examples of Hopf-Hecke algebras with a generalized Dirac cohomology. We show that they are in fact Barbasch-Sahi algebras, that is, a version of Vogan's conjecture analogous to the results of Huang and Pandžić in [J. Amer. Math. Soc., 2002] is available for them. We derive an explicit formula for the square of the Dirac operator and use it to study the finite-dimensional irreducible modules. We find that the Dirac cohomology of these modules is non-zero and that it, in fact, determines the modules uniquely.

math.RA↗

arXiv2020

Whittaker supports for representations of reductive groups

Let $F$ be either $\mathbb{R}$ or a finite extension of $\mathbb{Q}_p$, and let $G$ be a finite central extension of the group of $F$-points of a reductive group defined over $F$. Also let $π$ be a smooth representation of $G$ (Frechet of moderate growth if $F=\mathbb{R}$). For each nilpotent orbit $\mathcal{O}$ we consider a certain Whittaker quotient $π_{\mathcal{O}}$ of $π$. We define the Whittaker support WS$(π)$ to be the set of maximal $\mathcal{O}$ among those for which $π_{\mathcal{O}}\neq 0$. In this paper we prove that all $\mathcal{O}\in\mathrm{WS}(π)$ are quasi-admissible nilpotent orbits, generalizing some of the results in [Moe96,JLS16]. If $F$ is $p$-adic and $π$ is quasi-cuspidal then we show that all $\mathcal{O}\in\mathrm{WS}(π)$ are $F$-distinguished, i.e. do not intersect the Lie algebra of any proper Levi subgroup of $G$ defined over $F$. We also give an adaptation of our argument to automorphic representations, generalizing some results from [GRS03,Shen16,JLS16,Cai] and confirming some conjectures from [Ginz06]. Our methods are a synergy of the methods of the above-mentioned papers, and of our preceding paper [GGS17].

math.RT↗

arXiv2019

Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups

In this paper we analyze Fourier coefficients of automorphic forms on a finite cover $G$ of an adelic split simply-laced group. Let $π$ be a minimal or next-to-minimal automorphic representation of $G$. We prove that any $η\in π$ is completely determined by its Whittaker coefficients with respect to (possibly degenerate) characters of the unipotent radical of a fixed Borel subgroup, analogously to the Piatetski-Shapiro--Shalika formula for cusp forms on $GL_n$. We also derive explicit formulas expressing the form, as well as all its maximal parabolic Fourier coefficient in terms of these Whittaker coefficients. A consequence of our results is the non-existence of cusp forms in the minimal and next-to-minimal automorphic spectrum. We provide detailed examples for $G$ of type $D_5$ and $E_8$ with a view towards applications to scattering amplitudes in string theory.

math.NT↗

arXiv2019

Generalized and degenerate Whittaker quotients and Fourier coefficients

The study of Whittaker models for representations of reductive groups over local and global fields has become a central tool in representation theory and the theory of automorphic forms. However, only generic representations have Whittaker models. In order to encompass other representations, one attaches a degenerate (or a generalized) Whittaker model $W_{\mathcal{O}}$, or a Fourier coefficient in the global case, to any nilpotent orbit $\mathcal{O}$. In this note we survey some classical and some recent work in this direction - for Archimedean, p-adic and global fields. The main results concern the existence of models. For a representation $π$, call the set of maximal orbits $\mathcal{O}$ with $W_{\mathcal{O}}$ that includes $π$ the Whittaker support of $π$. The two main questions discussed in this note are: (1) What kind of orbits can appear in the Whittaker support of a representation? (2) How does the Whittaker support of a given representation $π$ relate to other invariants of $π$, such as its wave-front set?

math.RT↗

arXiv2019

Some remarks on non-symmetric interpolation Macdonald polynomials

We provide elementary identities relating the three known types of non-symmetric interpolation Macdonald polynomials. In addition we derive a duality for non-symmetric interpolation Macdonald polynomials. We consider some applications of these results, in particular for binomial formulas involving non-symmetric interpolation Macdonald polynomials.

math.QA↗

arXiv2019

Metaplectic representations of Hecke algebras, Weyl group actions, and associated polynomials

Chinta and Gunnells introduced a rather intricate multi-parameter Weyl group action on rational functions on a torus, which, when the parameters are specialized to certain Gauss sums, describes the functional equations of Weyl group multiple Dirichlet series associated to metaplectic (n-fold) covers of algebraic groups. In subsequent joint work with Puskas, they extended this action to a "metaplectic" representation of the equal parameter affine Hecke algebra, which allowed them to obtain explicit formulas for the p-parts of these Dirichlet series. They have also verified by a computer check the remarkable fact that their formulas continue to define a group action for general (unspecialized) parameters. In the first part of paper we give a conceptual explanation of this fact, by giving a uniform and elementary construction of the "metaplectic" representation for generic Hecke algebras as a suitable quotient of a parabolically induced affine Hecke algebra module, from which the associated Chinta-Gunnells Weyl group action follows through localization. In the second part of the paper we extend the metaplectic representation to the double affine Hecke algebra, which provides a generalization of Cherednik's basic representation. This allows us to introduce a new family of "metaplectic" polynomials, which generalize nonsymmetric Macdonald polynomials. In this paper, we provide the details of the construction of metaplectic polynomials in type A; the general case will be handled in the sequel to this paper.

math.RT↗

arXiv2019

The Capelli eigenvalue problem for Lie superalgebras

For a finite dimensional unital complex simple Jordan superalgebra $J$, the Tits-Kantor-Koecher construction yields a 3-graded Lie superalgebra $\mathfrak g_\flat\cong \mathfrak g_\flat(-1)\oplus\mathfrak g_\flat(0)\oplus\mathfrak g_\flat(1)$, such that $\mathfrak g_\flat(-1)\cong J$. Set $V:=\mathfrak g_\flat(-1)^*$ and $\mathfrak g:=\mathfrak g_\flat(0)$. In most cases, the space $\mathcal P(V)$ of superpolynomials on $V$ is a completely reducible and multiplicity-free representation of $\mathfrak g$, with a decomposition $\mathcal P(V):=\bigoplus_{λ\inΩ}V_λ$, where $\left(V_λ\right)_{λ\inΩ}$ is a family of irreducible $\mathfrak g$-modules parametrized by a set of partitions $Ω$. In these cases, one can define a natural basis $\left(D_λ\right)_{λ\inΩ}$ of "Capelli operators" for the algebra $\mathcal{PD}(V)^{\mathfrak g}$. In this paper we complete the solution to the Capelli eigenvalue problem, which is to determine the scalar $c_μ(λ)$ by which $D_μ$ acts on $V_λ$. We associate a restricted root system $\mathitΣ$ to the symmetric pair $(\mathfrak g,\mathfrak k)$ that corresponds to $J$, which is either a deformed root system of type $\mathsf{A}(m,n)$ or a root system of type $\mathsf{Q}(n)$. We prove a necessary and sufficient condition on the structure of $\mathitΣ$ for $\mathcal{P}(V)$ to be completely reducible and multiplicity-free. When $\mathitΣ$ satisfies the latter condition we obtain an explicit formula for the eigenvalue $c_μ(λ)$, in terms of Sergeev-Veselov's shifted super Jack polynomials when $\mathitΣ$ is of type $\mathsf{A}(m,n)$, and Okounkov-Ivanov's factorial Schur $Q$-polynomials when $\mathitΣ$ is of type $\mathsf{Q}(n)$.

math.RT↗

arXiv2018

A combinatorial formula for certain binomial coefficients for Jack polynomials

We present a decomposition of the generalized binomial coefficients associated with Jack polynomials into two factors: a stem, which is described explicitly in terms of hooks of the indexing partitions, and a leaf, which inherits various recurrence properties from the binomial coefficients and depends exclusively on the skew diagram. We then derive a direct combinatorial formula for the leaf in the special case where the two indexing partitions differ by at most two rows. This formula also exhibits an unexpected symmetry with respect to the lengths of the two rows.

math.CO↗

arXiv2018

Quadratic Capelli operators and Okounkov polynomials

Let $Z$ be the symmetric cone of $r \times r$ positive definite Hermitian matrices over a real division algebra $\mathbb F$. Then $Z$ admits a natural family of invariant differential operators -- the Capelli operators $C_λ$ -- indexed by partitions $λ$ of length at most $r$, whose eigenvalues are given by specialization of Knop--Sahi interpolation polynomials. In this paper we consider a double fibration $Y \longleftarrow X \longrightarrow Z$ where $Y$ is the Grassmanian of $r$-dimensional subspaces of $\mathbb F^n $ with $n \geq 2r$. Using this we construct a family of invariant differential operators $D_{λ,s}$ on $Y$ that we refer to as quadratic Capelli operators. Our main result shows that the eigenvalues of the $D_{λ,s}$ are given by specializations of Okounkov interpolation polynomials.

math.RT↗