Finkelberg–Mirković Schubert Scheme

• The Finkelberg–Mirković Schubert scheme is a scheme-theoretic model for spherical Schubert varieties in the affine Grassmannian, defined via representation-theoretic lattice containments.
• It employs an explicit tangent-space theorem to identify nilpotent thickenings and assess reducedness, notably revealing nonreduced structures in type E8.
• In type A, the scheme underpins noncommutative resolutions and the construction of canonical bases in equivariant K-theory through the Beilinson–Drinfeld fiber.

The Finkelberg–Mirković (FM) Schubert scheme is a scheme-theoretic model for spherical Schubert geometry in the affine Grassmannian. In its general form, it is a closed subscheme of the affine Grassmannian cut out by representation-theoretic lattice containments; its reduced subscheme is the corresponding spherical Schubert variety. In type $A$, the same program is realized through a special Beilinson–Drinfeld fiber $X_N$, which functions as a uniform base for colength-$N$ affine Schubert geometry and its convolution diagrams. Recent work has made this object central both in the study of reducedness and tangent spaces, and in the construction of noncommutative resolutions, perversely-exotic $t$-structures, and canonical bases in equivariant $K$-theory (Besson et al., 18 Mar 2026, Dumanski et al., 14 Jun 2026).

1. Modular definition and basic geometry

Let $G$ be a simply-connected simple algebraic group over $\mathbb C$. With $LG(R)=G(R((t)))$ and $L^+G(R)=G(R[[t]])$, the affine Grassmannian is the fpqc sheaf quotient

$\operatorname{Gr}_G = LG/L^+G.$

For a dominant coweight $X_N$0, the $X_N$1-orbit through $X_N$2 is the spherical Schubert cell $X_N$3, and its reduced closure is the spherical Schubert variety $X_N$4 (Besson et al., 18 Mar 2026).

The FM Schubert scheme $X_N$5 is defined by a lattice condition in every highest-weight representation. If an $X_N$6-point of $X_N$7 is represented by a pair $X_N$8, where $X_N$9 is a $N$0-torsor on $N$1 and $N$2 is a trivialization on $N$3, then for each dominant weight $N$4 one obtains a lattice

$N$5

The defining FM condition is

$N$6

This gives a representable closed subfunctor of $N$7. There is a canonical closed immersion

$N$8

and one has

$N$9

For simply-connected semisimple $t$0, the strengthened FM$t$1-condition of Kisin–Pappas–Zhou, namely the determinant constraint

$t$2

holds automatically on $t$3. Consequently,

$t$4

This identifies the FM Schubert scheme as the relevant scheme structure attached to the reduced Schubert support, rather than a different moduli problem.

2. Type $t$5 realization via the Beilinson–Drinfeld special fiber

For $t$6, with $t$7 and $t$8, the affine Grassmannian is

$t$9

equipped with a $K$0-action and loop rotation by $K$1. Schubert varieties admit a lattice-moduli description in $K$2: for dominant $K$3, the closure $K$4 parametrizes lattices $K$5 with prescribed relative position to the standard lattice $K$6 (Dumanski et al., 14 Jun 2026).

In the Finkelberg–Mirković program, a convenient scheme model for Schubert closures and convolution diagrams is provided by the special fiber of the Beilinson–Drinfeld Grassmannian. In type $K$7, the fundamental object is

$K$8

a projective $K$9-scheme carrying a natural $G$0-action. The paper explicitly identifies this $G$1 as the FM “Schubert scheme” in the sense that it is the scheme-theoretic receptacle for all colength-$G$2 stratified Schubert geometry on $G$3.

A decisive auxiliary object is the $G$4-torsor

$G$5

together with the canonical $G$6-equivariant map

$G$7

sending $G$8 to the endomorphism induced by $G$9 on $\mathbb C$0. Its image consists of matrices with at most $\mathbb C$1 Jordan blocks. This torsor and moment map provide the scheme-theoretic bridge between affine-Grassmannian coherent geometry and the nilpotent-cone geometry entering modified Lusztig correspondences.

The same type-$\mathbb C$2 framework produces the convolution or Bott–Samelson-type resolution. For a sequence of minuscule coweights $\mathbb C$3, with $\mathbb C$4 and $\mathbb C$5,

$\mathbb C$6

and there is a proper birational map

$\mathbb C$7

This resolution is smooth and projective over $\mathbb C$8, and $\mathbb C$9 is semismall for minuscule data.

3. Tangent-space theory and local structure at the base point

The local structure of FM Schubert schemes is controlled by an explicit tangent-space description. At the base point $LG(R)=G(R((t)))$0, Faltings’ open immersion $LG(R)=G(R((t)))$1 identifies

$LG(R)=G(R((t)))$2

as a $LG(R)=G(R((t)))$3-representation. This is the ambient linear space in which both reduced and nonreduced tangent directions are measured (Besson et al., 18 Mar 2026).

For the reduced spherical Schubert variety, a Demazure-module criterion gives

$LG(R)=G(R((t)))$4

In the quasi-minuscule case $LG(R)=G(R((t)))$5, one obtains

$LG(R)=G(R((t)))$6

The paper gives both a representation-theoretic proof, using the structure of the level-one highest-weight module $LG(R)=G(R((t)))$7, and a geometric proof via the identification of a neighborhood of $LG(R)=G(R((t)))$8 in $LG(R)=G(R((t)))$9 with the minimal nilpotent orbit closure $L^+G(R)=G(R[[t]])$0.

For the FM Schubert scheme, tangent vectors are described through dual numbers. Writing $L^+G(R)=G(R[[t]])$1, every tangent vector at $L^+G(R)=G(R[[t]])$2 can be represented as

$L^+G(R)=G(R[[t]])$3

and the FM condition becomes

$L^+G(R)=G(R[[t]])$4

The resulting tangent-space theorem is explicit. Define

$L^+G(R)=G(R[[t]])$5

Then

$L^+G(R)=G(R[[t]])$6

This formula isolates the nilpotent thickenings of the FM scheme directly in terms of the minimal dominant pairing data of $L^+G(R)=G(R[[t]])$7. A plausible implication is that the FM scheme records not only the Schubert support but also a representation-theoretically governed hierarchy of infinitesimal directions.

4. Reducedness, the quasi-minuscule case, and the $L^+G(R)=G(R[[t]])$8 counterexample

A central problem has been the reducedness conjecture: whether $L^+G(R)=G(R[[t]])$9 is always reduced, equivalently whether

$\operatorname{Gr}_G = LG/L^+G.$0

as schemes. The tangent-space formula for quasi-minuscule coweights resolves this question negatively in general (Besson et al., 18 Mar 2026).

For $\operatorname{Gr}_G = LG/L^+G.$1, the paper shows that if $\operatorname{Gr}_G = LG/L^+G.$2 is not of type $\operatorname{Gr}_G = LG/L^+G.$3, then $\operatorname{Gr}_G = LG/L^+G.$4, and hence

$\operatorname{Gr}_G = LG/L^+G.$5

Thus, in all simple simply-connected types other than $\operatorname{Gr}_G = LG/L^+G.$6, the FM tangent space at $\operatorname{Gr}_G = LG/L^+G.$7 matches the tangent space of the reduced spherical Schubert variety in the quasi-minuscule case.

In type $\operatorname{Gr}_G = LG/L^+G.$8, however,

$\operatorname{Gr}_G = LG/L^+G.$9

so $X_N$00. Therefore

$X_N$01

whereas

$X_N$02

The strict inclusion of tangent spaces detects nilpotent directions at the base point and proves that $X_N$03 is not reduced in type $X_N$04.

This gives a concrete counterexample to the FM reducedness conjecture. The same work notes that analogous nonreduced phenomena occur for truncated shifted Yangians. By contrast, the reducedness conjecture is known in type $X_N$05, so the $X_N$06 phenomenon is not generic across root systems but depends sensitively on the root-theoretic structure of the quasi-minuscule coroot.

5. Noncommutative resolutions and perversely-exotic $X_N$07-structures in type $X_N$08

For $X_N$09, the FM-Schubert-scheme framework supports a coherent and noncommutative refinement of affine Schubert geometry. Given the convolution resolution

$X_N$10

the paper constructs a $X_N$11-equivariant vector bundle $X_N$12 on $X_N$13, described as a BM-type tilting bundle whose restriction to every transverse slice is a tilting generator in the sense of Kaledin–Bezrukavnikov–Mirković. It defines the sheaf of algebras

$X_N$14

which is coherent and noncommutative on $X_N$15 (Dumanski et al., 14 Jun 2026).

Because $X_N$16 is a relative tilting generator, one obtains canonical derived equivalences

$X_N$17

and, equivariantly,

$X_N$18

This is the type-$X_N$19 affine-Grassmannian counterpart of the noncommutative Springer resolution.

The same pattern persists for Steinberg objects. For compatible $X_N$20, the affine-Grassmannian Steinberg variety is the derived fiber product

$X_N$21

and the endomorphism dg-algebra is

$X_N$22

All of these constructions work in characteristic $X_N$23 and over algebraically closed fields of sufficiently large positive characteristic.

Transporting the perverse-coherent $X_N$24-structure on $X_N$25 back across the equivalence yields the perversely-exotic $X_N$26-structure on

$X_N$27

Its heart is of finite length and is Artinian/Noetherian; standard and costandard objects are coherent IC-extensions from $X_N$28-orbits in $X_N$29, and simple objects are the corresponding perverse-coherent IC sheaves. Compatibility with transverse slices, including the Mirković–Vybornov slice isomorphism to parabolic Slodowy/MV slices, is built into the construction.

The small-rank examples in the paper illustrate the range of behavior. For $X_N$30 and $X_N$31, the Schubert variety is smooth, $X_N$32, $X_N$33 may be taken to be $X_N$34, and the perversely-exotic $X_N$35-structure agrees with the standard one. For $X_N$36 and $X_N$37, the resolution is genuinely nontrivial, and the perversely-exotic simples are IC-extensions from the $X_N$38-orbits inside $X_N$39.

6. Canonical bases, modified Lusztig correspondences, and the broader FM program

The type-$X_N$40 FM Schubert scheme $X_N$41 is also the base for a system of modified Lusztig correspondences. The $X_N$42-torsor $X_N$43 gives smooth $X_N$44-equivariant morphisms from affine-Grassmannian resolutions and Steinberg varieties to parabolic Springer and parabolic Springer–Steinberg varieties on $X_N$45 with the condition of at most $X_N$46 Jordan blocks. After a harmless $X_N$47-shift, these maps induce isomorphisms on equivariant $X_N$48-theory, both for resolutions and for Steinberg varieties (Dumanski et al., 14 Jun 2026).

This is the technical engine behind the main $X_N$49-theoretic theorem. Let $X_N$50 be the affine Hecke algebra of $X_N$51, and form the parabolic modules and anti-spherical modules with their Kazhdan–Lusztig canonical bases. After quotienting by basis elements attached to nilpotent orbits with at least $X_N$52 Jordan blocks, there are natural maps

$X_N$53

and

$X_N$54

These maps are injective for $X_N$55 and isomorphisms for $X_N$56. Under them, the Kazhdan–Lusztig canonical bases become the perversely-exotic bases given by classes of simple objects in the perversely-exotic heart.

The same package is related to quantum affine canonical bases via affine Schur algebras and skew Howe duality. There is a surjective algebra homomorphism

$X_N$57

compatible with canonical bases. Consequently,

$X_N$58

becomes a cyclic $X_N$59-module whose canonical basis is the perversely-exotic basis. Compatibility as $X_N$60 increases yields a limiting action in the sense of Cautis–Kamnitzer.

A complementary reduced-geometry manifestation of the FM viewpoint appears in the study of parabolic Mirković–Vilonen intersections. For a split reductive group $X_N$61, a standard parabolic $X_N$62, and reduced intersections

$X_N$63

there are explicit cellular pavings by affine spaces and tori indexed by positively folded alcove walks. Maximal-dimension walks parametrize top-dimensional irreducible components, and their cardinalities compute Levi-branching multiplicities $X_N$64 (Haines, 2024). This does not define the FM Schubert scheme itself, but it places FM-type affine-Schubert geometry in a concrete combinatorial framework for constant terms, branching, and convolution phenomena.

In this broader perspective, the FM Schubert scheme is both a local modular object and a global organizing device. Locally, it detects nilpotent directions beyond the reduced Schubert support; globally, especially in type $X_N$65, it furnishes the base over which resolutions, Steinberg correspondences, coherent–constructible compatibilities, canonical bases, and quantum affine actions can be transported from nilpotent-cone geometry to affine Schubert geometry.