arXiv subjects

Siddhartha Sahi

Publications and source records attributed to Siddhartha Sahi.

At least 19 recordsLinked to original sources

arXiv2026

On the Classification of Stein spaces with Bergman-Einstein metrics

For every $N\ge 2$, we prove that the Bergman metric on the regular locus of a finite ball quotient $\mathbb{B}^N/\Gamma$, where $\Gamma\subset \mathrm{U}(N)$ is finite and fixed-point-free, is K\"ahler-Einstein if and only if $\Gamma$ is trivial. Consequently, if $\Omega$ is an $N$-dimensional normal Stein space with isolated singularities and compact, smooth, strongly pseudoconvex boundary admitting a real-algebraic CR realization, then the Bergman metric on $\Omega_{\mathrm{reg}}$ is K\"ahler-Einstein if and only if $\Omega$ is biholomorphic to $\mathbb{B}^N$. This proves an algebraic version of the Cheng-Huang-Xiao conjecture in every complex dimension $N\ge 2$.

math.CV↗

arXiv2026

The Opdam-Cherednik kernel is the Laplace transform of a positive measure

We prove that the Opdam-Cherednik kernel, also known as the nonsymmetric Opdam hypergeometric function, can be written as the Laplace transform of a positive measure supported on the convex hull of the Weyl group orbit of its argument. As a consequence, the trigonometric Dunkl intertwining operator is positivity preserving. The main ingredient in the proof is a new formula for the Opdam-Cherednik kernel as a degeneration of nonsymmetric Macdonald polynomials. As a further application, we prove majorization inequalities for Macdonald polynomials and Heckman-Opdam hypergeometric functions associated with arbitrary root systems.

math.CA↗

arXiv2026

The Harish-Chandra isomorphism for supersymmetric spaces and ghost distributions

We prove the Harish-Chandra isomorphism theorem for supersymmetric spaces, describing the polynomial algebra of eigenvalues of invariant differential operators. The polynomials obtained satisfy novel invariance conditions, which remain somewhat mysterious. We also prove the Harish-Chandra isomorphism for ghost distributions, which satisfy a `square root' of the invariance conditions coming from invariant differential operators. All proofs are algebraic, and rely on a rank-one reduction argument and the Chevalley restriction theorem.

math.RT↗

arXiv2026

Majorization Inequalities from Logarithmic Convexity

Majorization inequalities for symmetric polynomials have interested mathematicians for centuries, from the AM-GM inequality for two variables going back at least to Baudh\={a}yana's \'{S}ulbas\={u}tra and Euclid's Elements in the first millenium BCE, through classical results of Newton, Muirhead and Gantmacher, to more recent extensions to Schur polynomials and zonal spherical functions. These have been established case by case, with no unified approach. Although it is known that majorization inequalities follow from symmetry and convexity in the indexing partition, the difficulty of proving convexity in specific cases has left a number of outstanding conjectures inaccessible until now. The key insight of this paper is that log-convexity provides a more versatile tool and a unifying principle. It implies convexity and hence majorization, and it is preserved under multiplication and weighted averaging, making it well suited to inductive arguments in a wide range of settings. Using this idea, we prove new majorization inequalities for Macdonald polynomials, Jack polynomials and Heckman-Opdam hypergeometric functions, unifying existing results and resolving several open conjectures.

math.CO↗

arXiv2026

Classification of irreducible real modules of real Lie superalgebras

We classify irreducible finite-dimensional modules of a collection of real Lie superalgebras that includes the simple ones, their classical variants, complex Lie superalgebras after restriction of scalars, and all real Lie algebras. Our strategy is to reduce this classification to determining the orbits of the parity and conjugation functors on irreducible modules of the complexifications of the aforementioned algebras. Then we provide explicit results for the computation of these orbits. For Lie superalgebras of basic type or of type $\mathbf Q(n)$, our classification applies to any highest-weight parametrization of irreducible complex modules with respect to an arbitrary Borel subalgebra. As a consequence, in the special case of real simple Lie algebras we obtain a new perspective on the classification of real simple modules and establish a conceptual connection with Kostant's cascade of strongly orthogonal roots.

math.RT↗

arXiv2025

A Characterization of Macdonald's Jack Hypergeometric Series ${}_pF_q(x;α)$ and ${}_pF_q(x,y;α)$ via Differential Equations

In a widely circulated manuscript from the 1980s, now available on the arXiv, I.~G.~Macdonald introduced certain multivariable hypergeometric series ${}_pF_q(x)= {}_pF_q(x;α)$ and ${}_pF_q(x,y)= {}_pF_q(x,y;α)$ in one and two sets of variables $x=(x_1,\dots x_n)$ and $y=(y_1,\dots y_n)$. These two series are defined by explicit expansions in terms of Jack polynomials $J^{(α)}_λ$, and for $α=2$ they specialize to the hypergeometric series of matrix arguments studied by Herz (1955) and Constantine (1963) that admit analogous expansions in terms of zonal polynomials. In this paper we determine explicit partial differential equations that characterize ${}_pF_q$, thereby answering a question posed by Macdonald. More precisely, for each $n,p,q$ we construct three differential operators $\mathcal A=\mathcal A^{(x,y)}$, $\mathcal B=\mathcal B^{(x)}$, $\mathcal C=C^{(x)}$, and we show that ${}_pF_q(x,y)$ and ${}_pF_q(x)$ are the unique series solutions of the equations $\mathcal A(f)=0$ and $\mathcal C(f)=0$, respectively, subject to certain symmetry and boundary conditions. We also prove that the equation $\mathcal B(f)=0$ characterizes ${}_pF_q(x)$, but only after one restricts the domain of $\mathcal B$ to the set of series satisfying an additional stability condition with respect to $n$. Special cases of the operators $\mathcal A$ and $\mathcal B$ have been constructed previously in the literature, but only for a small number of pairs $(p,q)$, namely for $p \leq 3$ and $q \leq 2$ in the zonal case by Muirhead (1970), Constantine--Muirhead (1972), and Fujikoshi (1975); and for $p \leq 2$ and $q \leq 1$ in the general Jack case by Macdonald (1980s), Yan (1992), Kaneko (1993), and Baker--Forrester (1997). However the operator $\mathcal C$ seems to be new even for these special cases.

math.CO↗

arXiv2025

Majorization via positivity of Jack and Macdonald polynomial differences

Majorization inequalities have a long history, going back to Maclaurin and Newton. They were recently studied for several families of symmetric functions, including by Cuttler--Greene--Skandera (2011), Sra (2016), Khare--Tao (2021), McSwiggen--Novak (2022), and Chen--Sahi (2024+) among others. Here we extend the inequalities by these authors to Jack and Macdonald polynomials, and obtain conjectural characterizations of majorization and of weak majorization of the underlying partitions. We prove these characterizations for two variables. In fact, we upgrade -- and prove in the above cases -- the characterization of majorization, to containment of Jack and Macdonald differences lying in the Muirhead semiring.

math.CO↗

arXiv2025

Monotonicity for generalized binomial coefficients and Jack positivity

Binomial formulas for Schur polynomials and Jack polynomials were studied by Lascoux in 1978, and Kaneko, Okounkov--Olshanski and Lassalle in the 1990s. We prove that the associated binomial coefficients are monotone and derive some symmetric function inequalities, in particular, a Schur positivity and Jack positivity result. These inequalities are similar to those studied by Newton, Muirhead, Gantmacher, Cuttler--Greene--Skandera, Sra and Khare--Tao.

math.CO↗

arXiv2025

Eigenvalues of supersymmetric Shimura operators and interpolation polynomials

The Shimura operators are a certain distinguished basis for invariant differential operators on a Hermitian symmetric space. Answering a question of Shimura, Sahi and Zhang showed that the Harish-Chandra images of these operators are specializations of certain $BC$-symmetric interpolation polynomials that were defined by Okounkov. We consider the analogs of Shimura operators for the Hermitian symmetric superpair $(\mathfrak{g},\mathfrak{k})$ where $\mathfrak{g}= \mathfrak{gl}(2p|2q)$ and $\mathfrak{k}= \mathfrak{gl}(p|q)\oplus \mathfrak{gl}(p|q)$ and we prove their Harish-Chandra images are specializations of certain $BC$-supersymmetric interpolation polynomials introduced by Sergeev--Veselov.

math.RT↗

arXiv2025

Simultaneous elections in a polarized society make single-party sweeps more likely

In a country with many elections, it may prove economically expedient to hold multiple elections simultaneously on a common polling date. We show that in a polarized society, in which each voter has a preferred party, an increase in the simultaneity of polling will increase the likelihood of a single-party sweep, namely, it will become more likely that a single party wins all the elections. In fact we show that the sweep probability goes up for \emph{every} party. Thus the phenomenon we describe is independent of the ``coattail'' or ``down-ballot'' effect of a popular leader. It is a \emph{systemic} and \emph{persistent} macroscopic political change, effected by a combination of political polarization and simultaneity of polling. Our result holds under fairly general conditions and is applicable to many common real-world electoral systems, including \emph{first-past-the-post} (most voters) and \emph{party list proportional representation} (most countries). In the course of our proof, we obtain a generalization of the well-known Harris correlation inequality.

math.PR↗

arXiv2025

Quasi-polynomial representations of double affine Hecke algebras

We introduce an explicit family of representations of the double affine Hecke algebra $\mathbb{H}$ acting on spaces of quasi-polynomials, defined in terms of truncated Demazure-Lusztig type operators. We show that these quasi-polynomial representations provide concrete realizations of a natural family of cyclic $Y$-parabolically induced $\mathbb{H}$-representations. We recover Cherednik's well-known polynomial representation as a special case. The quasi-polynomial representation gives rise to a family of commuting operators acting on spaces of quasi-polynomials. These generalize the Cherednik operators, which are fundamental in the study of Macdonald polynomials. We provide a detailed study of their joint eigenfunctions, which may be regarded as quasi-polynomial, multi-parametric generalizations of nonsymmetric Macdonald polynomials. We also introduce generalizations of symmetric Macdonald polynomials, which are invariant under a multi-parametric generalization of the standard Weyl group action. We connect our results to the representation theory of metaplectic covers of reductive groups over non-archimedean local fields. We introduce root system generalizations of the metaplectic polynomials from our previous work by taking a suitable restriction and reparametrization of the quasi-polynomial generalizations of Macdonald polynomials. We show that metaplectic Iwahori-Whittaker functions can be recovered by taking the Whittaker limit of these metaplectic polynomials.

math.RT↗

arXiv2024

Putting all eggs in one basket: some insights from a correlation inequality

We give examples of situations -- stochastic production, military tactics, corporate merger -- where it is beneficial to concentrate risk rather than to diversify it, that is, to put all eggs in one basket. Our examples admit a dual interpretation: as optimal strategies of a single player (the `principal') or, alternatively, as dominant strategies in a non-cooperative game with multiple players (the `agents'). The key mathematical result can be formulated in terms of a convolution structure on the set of increasing functions on a Boolean lattice (the lattice of subsets of a finite set). This generalizes the well-known Harris inequality from statistical physics and discrete mathematics; we give a simple self-contained proof of this result, and prove a further generalization based on the game-theoretic approach.

math.PR↗

arXiv2024

Restriction Theorems and Root Systems for Symmetric Superspaces

In this paper we consider those involutions $θ$ of a finite-dimensional Kac-Moody Lie superalgebra $\mathfrak g$, with associated decomposition $\mathfrak g=\mathfrak k\oplus\mathfrak p$, for which a Cartan subspace $\mathfrak a$ in $\mathfrak p_{\bar 0}$ is self-centralizing in $\mathfrak p$. For such $θ$ the restriction map $C_θ$ from $\mathfrak p$ to $\mathfrak a$ is injective on the algebra $P(\mathfrak p)^{\mathfrak k}$ of $\mathfrak k$-invariant polynomials on $\mathfrak p$. There are five infinite families and five exceptional cases of such involutions, and for each case we explicitly determine the structure of $P(\mathfrak p)^{\mathfrak k}$ by giving a complete set of generators for the image of $C_θ$. We also determine precisely when the restriction map $R_θ$ from $P(\mathfrak g)^{\mathfrak g}$ to $P(\mathfrak p)^{\mathfrak k}$ is surjective. Finally we introduce the notion of a generalized restricted root system, and show that in the present setting the $\mathfrak a$-roots $Δ(\mathfrak a,\mathfrak g)$ always form such a system.

math.RT↗

arXiv2024

The Capelli eigenvalue problem for quantum groups

We introduce and study quantum Capelli operators inside newly constructed quantum Weyl algebras associated to three families of symmetric pairs. Both the center of a particular quantized enveloping algebra and the Capelli operators act semisimply on the polynomial part of these quantum Weyl algebras. We show how to transfer well-known properties of the center arising from the theory of quantum symmetric pairs to the Capelli operators. Using this information, we provide a natural realization of Knop-Sahi interpolation polynomials as functions that produce eigenvalues for quantum Capelli operators.

math.QA↗

arXiv2024

Quantized Weyl algebras, the double centralizer property, and a new First Fundamental Theorem for $U_q(\mathfrak{gl}_n)$

Let $\mathcal P:=\mathcal P_{m\times n}$ denote the quantized coordinate ring of the space of $m\times n$ matrices, equipped with natural actions of the quantized enveloping algebras $U_q(\mathfrak{gl}_m)$ and $U_q(\mathfrak{gl}_n)$. Let $\mathcal L$ and $\mathcal R$ denote the images of $U_q(\mathfrak{gl}_m)$ and $U_q(\mathfrak{gl}_n)$ in $\mathrm{End}(\mathcal P)$, respectively. We define a $q$-analogue of the algebra of polynomial-coefficient differential operators inside $\mathrm{End}(\mathcal P)$, henceforth denoted by $\mathcal{PD}$, and we prove that $\mathcal L\cap \mathcal{PD}$ and $\mathcal{R}\cap \mathcal{PD}$ are mutual centralizers inside $\mathcal{PD}$. Using this, we establish a new First Fundamental Theorem of invariant theory for $U_q(\mathfrak{gl}_n)$. We also compute explicit formulas in terms of $q$-determinants for generators of the intersections with $\mathcal{PD}$ of the images of the Cartan subalgebras of $U_q(\mathfrak{gl}_m)$ and $U_q(\mathfrak{gl}_n)$.

math.QA↗

arXiv2024

Interpolation Polynomials, Binomial Coefficients, and Symmetric Function Inequalities

Interpolation polynomials were introduced by Knop--Sahi in type $A$, and Okounkov in type $BC$. They are inhomogeneous polynomials whose top terms are Jack and Macdonald polynomials. Thus the expansion coefficients for the product of two interpolation polynomials, known as Littlewood--Richardson coefficients, generalize the corresponding coefficients for Jack/Macdonald polynomials. Special values of interpolation polynomials, known as binomial coefficients, arise in the binomial type expansions of Jack/Macdonald polynomials and Koornwinder polynomials. We prove a number of results for interpolation polynomials and the associated coefficients. These include positivity and monotonicity results for binomial coefficients, partial positivity results for Littlewood--Richardson coefficients, and weighted sum formulas for both kinds of coefficients. As a special case of our results we obtain a new symmetric function inequality, which establishes a ``duality'' between Jack expansion positivity for symmetric functions, and the containment order on partitions, with respect to the shifted basis $\Omega_\lambda({\bf1}+x;\tau)$, where ${\bf1} =(1,\ldots,1)$ and $\Omega_\lambda(x;\tau)=P_\lambda(x;\tau)/P_\lambda({\bf1};\tau)$ is the normalized Jack polynomial. Our inequality can be seen as an analog of the inequalities of Cuttler--Greene--Skandera+Sra and Khare--Tao, which establish similar dualities between evaluation positivity on the positive orthant, and the dominance and weak dominance orders on partitions, with respect to the normalized Schur basis $\Omega_\lambda(x)=s_\lambda(x)/s_\lambda({\bf1})$ and its shifted version $\Omega_\lambda({\bf1}+x)$, respectively. In contrast to our result, the Jack versions of the two latter inequalities, although expected to hold, have not yet been proved.

math.CO↗

arXiv2024

On the classification of hypergeometric families of orthogonal polynomials on the real line

Several important families of orthogonal polynomials on the real line are called ``hypergeometric'' since they can be explicitly described in terms of some hypergeometric series $_pF_q$ that uses the degree $n$ of the polynomial as a parameter. It is natural to ask if one can classify all such families. Indeed many classification results have been obtained in this direction, but only under the additional assumption that the polynomials are eigenfunctions of some second order operator. In this paper we initiate a new approach to this classification. We propose a definition of an HG family that makes precise, but also generalizes, the notion of a ``hypergeometric'' family. Our main result is that there are exactly 10 types of orthogonal HG families, 8 from the well-known Askey scheme and 2 additional types of families that can be expressed in terms of Lommel polynomials. Our methods in this paper are algebraic. In particular, we classify a wider class of quasi-orthogonal HG families, and this classification is valid over an arbitrary field of characteristic zero. We also define a more general class of rational HG families and prove a structure theorem for quasi-orthogonal families in this class. We provide examples for such families, that are in particular new families of orthogonal polynomials of potential interest.

math.CA↗

arXiv2024

A Stone-von Neumann equivalence of categories for smooth representations of the Heisenberg group

The classical Stone-von Neuman theorem relates the irreducible unitary representations of the Heisenberg group $H_n$ to non-trivial unitary characters of its center $Z$, and plays a crucial role in the construction of the oscillator representation for the metaplectic group. In this paper we extend these ideas to non-unitary and non-irreducible representations, thereby obtaining an equivalence of categories between certain representations of $Z$ and those of $H_n$. Our main result is a smooth equivalence, which involves the fundamental ideas of du Cloux on differentiable representations and smooth imprimitivity systems for Nash groups. We show how to extend the oscillator representation to the smooth setting and give an application to degenerate Whittaker models for representations of reductive groups. We also include an algebraic equivalence, which can be regarded as a generalization of Kashiwara's lemma from the theory of $D$-modules.

math.RT↗