@article{21718,
  abstract     = {In this paper, we consider the big algebra recently introduced by Hausel for the GLn-action on the coordinate ring of the matrix space Mat(n,r). In particular, we obtain explicit formulas for the big algebra generators in terms of differential operators with polynomial coefficients. We show that big algebras in type A are commutative and relate them to the Bethe subalgebra in the Yangian Y(gln). We apply these results to big algebras of symmetric powers of the standard representation of GLn.
.},
  author       = {Ngo, Nhok T},
  issn         = {1815-0659},
  journal      = {Symmetry, Integrability and Geometry: Methods and Applications},
  publisher    = {National Academy of Science of Ukraine},
  title        = {{Big algebra in type A for the coordinate ring of the matrix space}},
  doi          = {10.3842/SIGMA.2026.024},
  volume       = {22},
  year         = {2026},
}

@article{21489,
  abstract     = {We study Kirillov algebras attached to minuscule highest weight representations of semisimple Lie algebras. They can be viewed as equivariant cohomology algebras of partial flag varieties. Real structures on the varieties then induce involutions of these algebras. We describe how these involutions act on the spectra of minuscule Kirillov algebras, and model the fixed points via the equivariant cohomology of real partial flag varieties. We then use this model to characterise freeness of the fixed point coordinate ring over the appropriate base. As an application, we recover a q = -1 phenomenon of Stembridge in the minuscule case by geometric means.},
  author       = {Elkner, Mischa M},
  issn         = {1531-586X},
  journal      = {Transformation Groups},
  publisher    = {Springer Nature},
  title        = {{On involutions of minuscule Kirillov algebras induced by real structures}},
  doi          = {10.1007/s00031-026-09958-y},
  year         = {2026},
}

@phdthesis{22694,
  abstract     = {We develop and employ techniques from equivariant algebraic K-theory and related invariants
in the context of geometric representation theory, in both arithmetic and topological situations.
We showcase the use of such techniques on the affine Grassmannian Gr, a space of fundamental
interest in the geometric Langlands program.

It is a deep development of mathematics of the last century that many concrete, yet combina-
torially complex algebraic problems may be effectively studied through the lens of algebraic
geometry. The objects of interest can be often realized as cohomological invariants of algebraic
varieties, and good understanding of their geometry sheds light into the original questions.
Such techniques have seen immense applications in the Langlands program, where they go
under the label of geometric representation theory.

One source of powerful invariants in algebraic geometry comes from algebraic K-theory,
Hochschild homology, and their relatives. These localizing invariants contain large amount
of information, but are quite hard to compute. For this reason, their usage in geometric
representation theory has been limited.

The aim of this thesis is to showcase how to control such invariants in the situations of
interest and use them to obtain new insights. We start by reinterpreting equivariant Hochschild
homology in terms of functions on certain fixed-point schemes, which are of independent
interest. We compare it to equivariant K-theory via the trace map. We give new computations
and comparisons of such invariants of affine Schubert varieties in Gr, including arithmetic
situations. We show that they behave much better than expected.

We finally utilize this circle of ideas in a purely topological setting. We describe the varying
fixed points of the extended torus action on the affine Grassmannian, and use it to compute
its equivariant topological K-theory ring. The answer is nontrivial and verifies an outstanding
conjecture in the subject.

We compare, partly conjecturally, the resulting K-theory ring to the completed center of an
integral even hybrid quantum group and its deformed quantum category O. This gives a
genuine application of our computations in pure representation theory.},
  author       = {Löwit, Jakub},
  issn         = {2663-337X},
  pages        = {185},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Equivariant K-theory of affine Grassmannians in representation theory and arithmetic}},
  doi          = {10.15479/AT-ISTA-22694},
  year         = {2026},
}

@article{22693,
  abstract     = {We study torus-equivariant algebraic K-theory of affine Schubert varieties in the perfect affine Grassmannians over Fp. We further compare it to the torus-equivariant Hochschild homology of perfect complexes, which has a geometric description in terms of global functions on certain fixed-point schemes. We prove that Fp-linearly, this comparison is an isomorphism. Our approach is quite constructive, resulting in new computations of these K-theory rings. We establish various structural results for equivariant perfect algebraic K-theory on the way; we believe these are of independent interest.},
  author       = {Löwit, Jakub},
  issn         = {1431-0643},
  journal      = {Documenta Mathematica},
  keywords     = {equivariant algebraic K-theory, perfection in positive characteristic, affine Grassmannian, affine Schubert varieties, Dennis trace map, equivariant Hochschild homology, fixed-point schemes, toric varieties},
  publisher    = {EMS Press},
  title        = {{Equivariant K-theory, affine Grassmannian and perfection}},
  doi          = {10.4171/dm/1064},
  year         = {2026},
}

@article{19071,
  abstract     = {An action of a complex reductive group G on a smooth projective variety X is regular when all regular unipotent elements in G act with finitely many fixed points. Then the complex G
-equivariant cohomology ring of X is isomorphic to the coordinate ring of a certain regular fixed point scheme. Examples include partial flag varieties, smooth Schubert varieties and Bott-Samelson varieties. We also show that a more general version of the fixed point scheme allows a generalisation to GKM spaces, such as toric varieties.},
  author       = {Hausel, Tamás and Rychlewicz, Kamil P},
  issn         = {2491-6765},
  journal      = {Epijournal de Geometrie Algebrique},
  publisher    = {EPI Sciences},
  title        = {{Spectrum of equivariant cohomology as a fixed point scheme}},
  doi          = {10.46298/epiga.2025.12591},
  volume       = {9},
  year         = {2025},
}

@article{19621,
  abstract     = {In this paper we obtain a complete description of all indecomposable characters (central positive-definite functions) of inductive limits of the symmetric groups under block diagonal embedding. As a corollary we obtain the full classification of the isomorphism classes of these inductive limits.},
  author       = {Nessonov, Nikolay and Ngo, Nhok T},
  issn         = {1088-4165},
  journal      = {Representation Theory},
  number       = {8},
  pages        = {256--288},
  publisher    = {American Mathematical Society},
  title        = {{Indecomposable characters of inductive limits of symmetric groups}},
  doi          = {10.1090/ert/689},
  volume       = {29},
  year         = {2025},
}

@article{18108,
  abstract     = {Here we announce the construction and properties of a big commutative subalgebra of the Kirillov algebra attached to a finite dimensional irreducible representation of a complex semisimple Lie group. They are commutative finite flat algebras over the cohomology of the classifying space of the group. They are isomorphic with the equivariant intersection cohomology of affine Schubert varieties, endowing the latter with a new ring structure. Study of the finer aspects of the structure of the big algebras will also furnish the stalks of the intersection cohomology with ring structure, thus ringifying Lusztig’s q-weight multiplicity polynomials i.e., certain affine Kazhdan–Lusztig polynomials.},
  author       = {Hausel, Tamás},
  issn         = {1091-6490},
  journal      = {Proceedings of the National Academy of Sciences of the United States of America},
  number       = {38},
  publisher    = {National Academy of Sciences},
  title        = {{Commutative avatars of representations of semisimple Lie groups}},
  doi          = {10.1073/pnas.2319341121},
  volume       = {121},
  year         = {2024},
}

@article{15339,
  abstract     = {We define even very stable Higgs bundles and study the Hitchin map restricted to their upward flows. In the GLn case, we classify the type (1,…,1) examples, and find that they are governed by a root system formed by the roots of even height. We discuss how the spectrum of equivariant cohomology of real and quaternionic Grassmannians, 4n-spheres and the real Cayley plane appear to describe the Hitchin map on even cominuscule upward flows. The even upward flows in question are the same as upward flows in Higgs bundle moduli spaces for quasi-split inner real forms. The latter spaces have been pioneered by Oscar García-Prada and his collaborators.},
  author       = {González, Miguel and Hausel, Tamás},
  issn         = {1793-6519},
  journal      = {International Journal of Mathematics},
  number       = {09},
  publisher    = {World Scientific Publishing},
  title        = {{Hitchin map on even very stable upward flows}},
  doi          = {10.1142/S0129167X2441009X},
  volume       = {35},
  year         = {2024},
}