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 ex
tended 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.
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 Macdonalds 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.
Let $(V,omega)$ be an orthosympectic $mathbb Z_2$-graded vector space and let $mathfrak g:=mathfrak{gosp}(V,omega)$ denote the Lie superalgebra of similitudes of $(V,omega)$. When the space $mathscr P(V)$ of superpolynomials on $V$ is emph{not} a com
pletely reducible $mathfrak g$-module, we construct a natural basis $D_lambda$ 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 $D_lambda$ 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 $D_lambda$ 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 ${mathbf D_{t,lambda}}_{lambdainmathcal P}^{}$ 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 $left{mathbf D_{t,lambda}right}_{lambdainmathcal P}$ that are analogous to our results for the operators ${D_lambda}_{lambdainmathcal P}^{}$.
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 result
s, in particular for binomial formulas involving non-symmetric interpolation Macdonald 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 Dirichle
t 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 Cheredniks 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.
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 $pi$, call the set of maximal orbits $mathcal{O}$ with $W_{mathcal{O}}$ that includes $pi$ the Whittaker support of $pi$. 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 $pi$ relate to other invariants of $pi$, such as its wave-front set?
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 p
roperties 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.
For a finite dimensional unital complex simple Jordan superalgebra $J$, the Tits-Kantor-Koecher construction yields a 3-graded Lie superalgebra $mathfrak g_flatcong mathfrak g_flat(-1)oplusmathfrak g_flat(0)oplusmathfrak g_flat(1)$, such that $mathfr
ak g_flat(-1)cong J$. 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 $mathcal P(V):=bigoplus_{lambdainOmega}V_lambda$, where $left(V_lambdaright)_{lambdainOmega}$ is a family of irreducible $mathfrak g$-modules parametrized by a set of partitions $Omega$. In these cases, one can define a natural basis $left(D_lambdaright)_{lambdainOmega}$ 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 $c_mu(lambda)$ by which $D_mu$ acts on $V_lambda$. We associate a restricted root system $mathit{Sigma}$ 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 $mathit{Sigma}$ for $mathcal{P}(V)$ to be completely reducible and multiplicity-free. When $mathit{Sigma}$ satisfies the latter condition we obtain an explicit formula for the eigenvalue $c_mu(lambda)$, in terms of Sergeev-Veselovs shifted super Jack polynomials when $mathit{Sigma}$ is of type $mathsf{A}(m,n)$, and Okounkov-Ivanovs factorial Schur $Q$-polynomials when $mathit{Sigma}$ is of type $mathsf{Q}(n)$.
Let $mathfrak l:= mathfrak q(n)timesmathfrak q(n)$, where $mathfrak q(n)$ denotes the queer Lie superalgebra. The associative superalgebra $V$ of type $Q(n)$ has a left and right action of $mathfrak q(n)$, and hence is equipped with a canonical $math
frak l$-module structure. We consider a distinguished basis ${D_lambda}$ of the algebra of $mathfrak l$-invariant super-polynomial differential operators on $V$, which is indexed by strict partitions of length at most $n$. We show that the spectrum of the operator $D_lambda$, when it acts on the algebra $mathscr P(V)$ of super-polynomials on $V$, is given by the factorial Schur $Q$-function of Okounkov and Ivanov. This constitutes a refinement and a new proof of a result of Nazarov, who computed the top-degree homogeneous part of the Harish-Chandra image of $D_lambda$. As a further application, we show that the radial projections of the spherical super-polynomials corresponding to the diagonal symmetric pair $(mathfrak l,mathfrak m)$, where $mathfrak m:=mathfrak q(n)$, of irreducible $mathfrak l$-submodules of $mathscr P(V)$ are the classical Schur $Q$-functions.
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 $pi$ be a smooth representation of $G$ (Frechet of moderate gro
wth if $F=mathbb{R}$). For each nilpotent orbit $mathcal{O}$ we consider a certain Whittaker quotient $pi_{mathcal{O}}$ of $pi$. We define the Whittaker support WS$(pi)$ to be the set of maximal $mathcal{O}$ among those for which $pi_{mathcal{O}} eq 0$. In this paper we prove that all $mathcal{O}inmathrm{WS}(pi)$ are quasi-admissible nilpotent orbits, generalizing some of the results in [Moe96,JLS16]. If $F$ is $p$-adic and $pi$ is quasi-cuspidal then we show that all $mathcal{O}inmathrm{WS}(pi)$ 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].
