Pieri Integral Formula and Asymptotics of Jack Unitary Characters

We introduce Jack (unitary) characters and prove two kinds of formulas that are suitable for their asymptotics, as the lengths of the signatures that parametrize them go to infinity. The first kind includes several integral representations for Jack characters of one variable. The second identity we prove is the Pieri integral formula for Jack characters which, in a sense, is dual to the well known Pieri rule for Jack polynomials. The Pieri integral formula can also be seen as a functional equation for irreducible spherical functions of virtual Gelfand pairs. As an application of our formulas, we study the asymptotics of Jack characters as the corresponding signatures grow to infinity in the sense of Vershik-Kerov. We prove the existence of a small $\delta > 0$ such that the Jack characters of $m$ variables have a uniform limit on the $\delta$-neighborhood of the $m$-dimensional torus. Our result specializes to a theorem of Okounkov and Olshanski.Comment: 39 pages. v2: revised after the referee's comments. To appear in Selecta Mathematica, New Serie

q-Deformed character theory for infinite-dimensional symplectic and orthogonal groups

The classification of irreducible, spherical characters of the infinite-dimensional unitary/orthogonal/symplectic groups can be obtained by finding all possible limits of normalized, irreducible characters of the corresponding finite-dimensional groups, as the rank tends to infinity. We solve a q-deformed version of the latter problem for orthogonal and symplectic groups, extending previously known results for the unitary group. The proof is based on novel determinantal and double-contour integral formulas for the q-specialized characters

Infinite-dimensional groups over finite fields and Hall-Littlewood symmetric functions

The groups mentioned in the title are certain matrix groups of infinite size over a finite field $\mathbb F_q$. They are built from finite classical groups and at the same time they are similar to reductive $p$-adic Lie groups. In the present paper, we initiate the study of invariant measures for the coadjoint action of these infinite-dimensional groups. We examine first the group $\mathbb{GLB}$, a topological completion of the inductive limit group $\varinjlim GL(n, \mathbb F_q)$. As was shown by Gorin, Kerov, and Vershik [arXiv:1209.4945], the traceable factor representations of $\mathbb{GLB}$ admit a complete classification, achieved in terms of harmonic functions on the Young graph $\mathbb Y$. We show that there exists a parallel theory for ergodic coadjoint-invariant measures, which is linked with a deformed version of harmonic functions on $\mathbb Y$. Here the deformation means that the edges of $\mathbb Y$ are endowed with certain formal multiplicities coming from the simplest version of Pieri rule (multiplication by the first power sum $p_1$) for the Hall-Littlewood (HL) symmetric functions with parameter $t:=q^{-1}$. This fact serves as a prelude to our main results, which concern topological completions of two inductive limit groups built from finite unitary groups. We show that in this case, coadjoint-invariant measures are linked to some new branching graphs. The latter are still related to the HL functions, but the novelty is that now the formal edge multiplicities come from the multiplication by $p_2$ (not $p_1$) and the HL parameter $t$ turns out to be negative (as in Ennola's duality).Comment: 46 p

Universality of global asymptotics of Jack-deformed random Young diagrams at varying temperatures

This paper establishes universal formulas describing the global asymptotics of two different models of discrete $\beta$-ensembles in high, low and fixed temperature regimes. Our results affirmatively answer a question posed by the second author and \'Sniady. We first consider the Jack measures on Young diagrams of arbitrary size, which depend on the inverse temperature parameter $\beta>0$ and specialize to Schur measures when $\beta=2$. We introduce a class of Jack measures of Plancherel-type and prove a law of large numbers and central limit theorem in the three regimes. In each regime, we provide explicit formulas for polynomial observables of the limit shape and Gaussian fluctuations around the limit shape. These formulas have surprising positivity properties and are expressed in terms of weighted lattice paths. We also establish connections between these measures and the work of Kerov-Okounkov-Olshanski on Jack-positive specializations and show that this is a rich class of measures parametrized by the elements in the Thoma cone. Second, we show that the formulas from limits of Plancherel-type Jack measures are universal: they also describe the limit shape and Gaussian fluctuations for the second model of random Young diagrams of a fixed size defined by Jack characters with the approximate factorization property (AFP) studied by the second author and \'Sniady. Finally, we discuss the limit shape in the high/low-temperature regimes and show that, contrary to the continuous case of $\beta$-ensembles, there is a phase transition phenomenon in passing from the fixed temperature regime to the high/low temperature regimes. We note that the relation we find between the two different models of random Young diagrams appears to be new, even in the special case of $\beta=2$ that relates Schur measures to the character measures with the AFP studied by Biane and \'Sniady.Comment: 64 pages, 7 figure

The Elliptic Tail Kernel

We introduce and study a new family of $q$-translation-invariant determinantal point processes on the two-sided $q$-lattice. We prove that these processes are limits of the $q$-$zw$ measures, which arise in the $q$-deformation of harmonic analysis on $U(\infty)$, and express their correlation kernels in terms of Jacobi theta functions. As an application, we show that the $q$-$zw$ measures are diffuse. Our results also hint at a link between the two-sided $q$-lattice and rows/columns of Young diagrams.Comment: 28 page