We present a new combinatorial formula for Hall–Littlewood functions associated with the affine root system of type (Formula presented.), i.e., corresponding to the affine Lie algebra (Formula presented.). Our formula has the form of a sum over the elements of a basis constructed by Feigin, Jimbo, Loktev, Miwa and Mukhin in the corresponding irreducible representation. Our formula can be viewed as a weighted sum of exponentials of integer points in a certain infinite-dimensional convex polyhedron. We derive a weighted version of Brion’s theorem and then apply it to our polyhedron to prove the formula. © 2016 Springer International Publishing

A knot space in a manifold *M* is a space of oriented immersions *S*^{1} ↪ *M* up to Diff(*S* ^{1}). J.-L. Brylinski has shown that a knot space of a Riemannian threefold is formally Kähler. We prove that a space of knots in a holonomy *G* _{2} manifold is formally Kähler.

A theorem of Y. Berest, P. Etingof and V. Ginzburg states that finite-dimensional irreducible representations of a type A rational Cherednik algebra are classified by one rational number *m*/*n*. Every such representation is a representation of the symmetric group *S**n* . We compare certain multiplicity spaces in its decomposition into irreducible representations of *S**n* with the spaces of differential forms on a zero-dimensional moduli space associated with the plane curve singularity *x^**m*= *y*^*n* .

For a left coherent ring A with every left ideal having a countable set of generators, we show that the coderived category of left A-modules is compactly generated by the bounded derived category of finitely presented left A-modules (reproducing a particular case of a recent result of Št’ovíček with our methods). Furthermore, we present the definition of a dualizing complex of fp-injective modules over a pair of noncommutative coherent rings A and B, and construct an equivalence between the coderived category of A-modules and the contraderived category of B-modules. Finally, we define the notion of a relative dualizing complex of bimodules for a pair of noncommutative ring homomorphisms (Formula presented.) and (Formula presented.), and obtain an equivalence between the R / A-semicoderived category of R-modules and the S / B-semicontraderived category of S-modules. For a homomorphism of commutative rings (Formula presented.), we also construct a tensor structure on the R / A-semicoderived category of R-modules. A vision of semi-infinite algebraic geometry is discussed in the introduction.

In 1998, Leclerc and Zelevinsky introduced the notion of *weakly separated* collections of subsets of the ordered *n*-element set [*n*] (using this notion to give a combinatorial characterization for quasi-commuting minors of a quantum matrix). They conjectured the purity of certain natural domains D⊆2[n]D⊆2[n] (in particular, of the hypercube 2[n]2[n] itself, and the hyper-simplex {X⊆[n]:|X|=m}{X⊆[n]:|X|=m} for *m* fixed), where DD is called *pure* if all maximal weakly separated collections in DD have the same cardinality. These conjectures have been answered affirmatively. In this paper, generalizing those earlier results, we reveal wider classes of pure domains in 2[n]2[n]. This is obtained as a consequence of our study of a novel geometric–combinatorial model for weakly separated set-systems, so-called *combined* (*polygonal*) *tilings*on a zonogon, which yields a new insight in the area.

Given a variety Y with a rectangular Lefschetz decomposition of its derived category, we consider a degree n cyclic cover X→Y ramified over a divisor Z⊂Y. We construct semiorthogonal decompositions of Db(X) and Db(Z) with distinguished components AX and AZ and prove the equivariant category of AX (with respect to an action of the nth roots of unity) admits a semiorthogonal decomposition into n−1n−1 copies of AZ. As examples, we consider quartic double solids, Gushel–Mukai varieties, and cyclic cubic hypersurfaces.

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of *GL _{n}.* We calculate the equivariant cohomology rings of the Laumon moduli spaces in terms of Gelfand-Tsetlin subalgebra of

*U*(

*gl*), and formulate a conjectural answer for the small quantum cohomology rings in terms of certain commutative shift of argument subalgebras of

_{n}*U*(

*gl*).

_{n}Classical local Weyl modules for a simple Lie algebra are labeled by dominant weights. We generalize the definition to the case of arbitrary weights and study the properties of the generalized modules. We prove that the representation theory of the generalized Weyl modules can be described in terms of the alcove paths and the quantum Bruhat graph. We make use of the Orr–Shimozono formula in order to prove that the $t=\infty$ specializations of the nonsymmetric Macdonald polynomials are equal to the characters of certain generalized Weyl modules.

We discuss a conjecture saying that derived equivalence of simply connected smooth projective varieties implies that the difference of their classes in the Grothendieck ring of varieties is annihilated by a power of the affine line class. We support the conjecture with a number of known examples, and one new example. We consider a smooth complete intersection X of three quadrics in P5 and the corresponding double cover Y→P2 branched over a sextic curve. We show that as soon as the natural Brauer class on Y vanishes, so that X and Y are derived equivalent, the difference [X]−[Y] is annihilated by the affine line class.

Let *F*_{λ} be a generalized flag variety of a simple Lie group *G* embedded into the projectivization of an irreducible *G*-module *V*_{λ}. We define a flat degeneration *F*_{λ}^{a}, which is a *G*_{a}^{M} variety. Moreover, there exists a larger group *G*^{a} acting on *F*_{λ}^{a}, which is a degeneration of the group *G*. The group *G*^{a} contains *G*_{a}^{M} as a normal subgroup. If *G* is of type *A*, then the degenerate flag varieties can be embedded into the product of Grassmannians and thus to the product of projective spaces. The defining ideal of *F*_{λ}^{a} is generated by the set of degenerate Plüker relations. We prove that the coordinate ring of *F*_{λ}^{a} is isomorphic to a direct sum of dual PBW-graded *g*-modules. We also prove that there exist bases in multi-homogeneous components of the coordinate rings, parametrized by the semistandard PBW-tableux, which are analogues of semistandard tableaux.

We study the Maulik-Okounkov K-theoretic stable basis for the Hilbert scheme of points on the plane, and its dependence of the slope parameter.

We study the natural Gieseker and Uhlenbeck compactifications of the rational Calogero–Moser phase space. The Gieseker compactification is smooth and provides a small resolution of the Uhlenbeck compactification. We use the resolution to compute the stalks of the IC-sheaf of the Uhlenbeck compactification.

We study the category of graded representations with finite--dimensional graded pieces for the current algebra associated to a simple Lie algebra. This category has many similarities with the category O of modules for g and in this paper, we use the combinatorics of Macdonald polynomials to prove an analogue of the famous BGG duality in the case of sl(n+1)

We compute the Frobenius trace functions of mirabolic character sheaves defined over a finite field. The answer is given in terms of the character values of general linear groups over the finite field, and the structure constants of multiplication in the mirabolic Hall–Littlewood basis of symmetric functions, introduced by Shoji.

For a complex reflection group *W* with reflection representation hh, we define and study a natural filtration by Serre subcategories of the category O_*c*(*W*,h) of representations of the rational Cherednik algebra *H_**c*(*W*,h). This filtration refines the filtration by supports and is analogous to the Harish-Chandra series appearing in the representation theory of finite groups of Lie type. Using the monodromy of the Bezrukavnikov–Etingof parabolic restriction functors, we show that the subquotients of this filtration are equivalent to categories of finite-dimensional representations over generalized Hecke algebras. When *W* is a finite Coxeter group, we give a method for producing explicit presentations of these generalized Hecke algebras in terms of finite-type Iwahori–Hecke algebras. This yields a method for counting the number of irreducible objects in O_*c*(*W*,h) of given support. We apply these techniques to count the number of irreducible representations in O_*c*(*W*,h) of given support for all exceptional Coxeter groups *W* and all parameters *c*, including the unequal parameter case. This completes the classification of the finite-dimensional irreducible representations of O_*c*(*W*,h) for exceptional Coxeter groups *W* in many new cases.