We construct a new family of graded representations W lambda for the positive elliptic Hall algebra E+ indexed by Young diagrams lambda which generalize the standard E+ action on symmetric functions. These representations have homogeneous bases of eigenvectors for the action of the Macdonald element P-0,P-1 is an element of E+ with distinct Q(q, t)-rational spectrum generalizing the symmetric Macdonald functions. The analysis of the structure of these representations exhibits interesting combinatorics arising from the limits of periodic standard Young tableaux. We find an explicit combinatorial rule for the action of the multiplication operators e(r)(center dot) generalizing the Pieri rule for symmetric Macdonald functions.
We analyze three related families of q-series arising from characters of vertex (super)algebras. More precisely, we consider: (i) graph series and characters of principal subspaces associated with arc algebras; (ii) characters, or indices, of class-S vertex operator algebras in the context of 4d N = 2 SOFT; and (iii) supercharacters of the U family of vertex superalgebras and their quasi-modular properties. Along the way, we introduce a family of multiple q-zeta values associated with simple Lie algebras and present their conjectural properties.
We investigate the irreducible smooth bsl2-modules recently constructed in [Adv. Math. 481 (2025), 110559, 34 pages, arXiv:2404.03855], and demonstrate that these modules admit a Wakimoto-type realization at both critical and non-critical levels. In the critical level case, we identify simple quotients of these modules with the Wakimoto modules whose irreducibility was already established by Adamovic. We also generalize some Wakimoto modules constructed in [Adv. Math. 289 (2016), 438-479, arXiv:1409.5354] and identify them as generalized Whittaker modules.
Let M-p, be a circle bundle with first Chern class p[w] over a closed 4n-dimensional integral symplectic manifold (M-,w). Equivalently, M-p, is a closed contact (4n + 1)-manifold whose Reeb orbits are all closed and have the same period. For a metric gon M-p, compati ble with the symplectic structure and the geometry of the circle fiber, we use Wodzicki-Chern Simons forms on the loop space LMp, to prove that pi(1)(M-p,g)) is infinite for |p| >> 0 We also give the first high-dimensional examples of nonvanishing Wodzicki Pontryagin forms.
We conclude our work on asymptotic expansions at the soft edge for the classical $n$-dimensional Gaussian and Laguerre ensembles, now studying the gap-probability generating functions. We show that the correction terms in the asymptotic expansion are multilinear forms of the higher-order derivatives of the leading-order term, with certain rational polynomial coefficients that are independent of the dummy generating function variable. In this way, the same multilinear structure, with the same polynomial coefficients, is inherited by the asymptotic expansion of any linearly induced quantity such as the distribution of the $k$-th largest level. Whereas the results for the unitary ensembles are presented with proof, the discussion of the orthogonal and symplectic ones is based on some hypotheses. To substantiate the hypotheses, we check the result for the $k$-th largest level in the orthogonal ensembles against simulation data for choices of $n$ and $k$ that require as many as four correction terms to achieve satisfactory accuracy.
