Partial Springer Resolutions in Lie Theory

• Partial Springer resolutions are geometric constructions that generalize the classical Springer resolution by replacing Borel subgroups with parabolic ones, linking Lie theory and algebraic combinatorics.
• They reveal rich symplectic, Poisson, and topological structures by encoding singularities of nilpotent cones and supporting structures such as Weyl group actions and moment maps.
• Applications span representation theory, topological quantum field theory, and Hecke algebra analysis, providing practical tools for studying singular spaces and deformation theories.

A partial Springer resolution is a geometric and representation-theoretic construction that interpolates between the classical Springer resolution of the nilpotent cone and various natural singular quotient spaces arising in Lie theory, symplectic geometry, and algebraic combinatorics. These constructions play central roles in the topology of singular spaces (notably nilpotent cones and symplectic singularities), categorical representation theory, topological quantum field theory (TQFT), and the study of Hecke algebras and their central elements. Partial Springer resolutions are parameterized by parabolic subgroups or Levi data and frequently encode parabolic or symmetric structures, with intricate connections to Weyl groups, Poisson geometry, and enumerative combinatorics.

1. Classical and Partial Springer Resolutions: Framework

Let $G$ be a connected reductive group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}$. The classical Springer resolution is defined as the cotangent bundle $T^*(G/B)$ of the flag variety, mapping to the nilpotent cone $\mathcal{N}\subset\mathfrak{g}$ by sending a pair $(X,gB)$ to $X$, with $X$ nilpotent and $gB$ a Borel position such that $g^{-1} X\in\mathfrak{b}$. This is a symplectic resolution and has deep ties to representation theory via the Springer correspondence.

Partial Springer resolutions generalize this setting by replacing the Borel subgroup $B$ with a parabolic subgroup $P$ (with Levi factor $L$ and unipotent radical $U$). The main example is

$$\widetilde{\mathcal{N}}_P := G \times^P \mathcal{N}_L, \quad \pi_P: \widetilde{\mathcal{N}}_P \to \mathcal{N}$$

where $\mathcal{N}_L$ is the nilpotent cone in $\operatorname{Lie}(L)$. The map $\pi_P([g,x])=Ad(g)x$ is proper, $G$-equivariant, and birational to its image. The base $G/P$ is a partial flag variety.

Partial Springer resolutions can be constructed for incidence varieties associated with parabolics in $\mathfrak{g}$ or $G$, for example:

• For a conjugacy class $\mathcal{C}$ of parabolic subalgebras of $\mathfrak{g}$, one considers

$$\mathfrak{g}_\mathcal{C} = \{ (p, x) \in \mathcal{C} \times \mathfrak{g} : x \in p \}$$

and its associated moment map, forming a resolution $$\mu_\mathcal{C} : \mathfrak{g}_\mathcal{C} \to \mathfrak{g}$$ (Crooks et al., 14 Apr 2025).

• Parabolic analogs in the group context involve $$G_\mathcal{C} = \{ (P, g) \in \mathcal{C}\times G : g\in P \}$$ and the associated group-valued moment map $\nu_\mathcal{C}$ (Crooks et al., 14 Apr 2025).

2. Geometric and Poisson Structures

Partial Springer resolutions naturally inherit rich geometric and symplectic structures:

• The cotangent bundle $T^*G$ is a symplectic groupoid over $\mathfrak{g}$; its reduction along parabolic data yields subgroupoids $(T^*G)_\mathcal{C}$ integrating induced Poisson structures on $\mathfrak{g}_\mathcal{C}$ (Crooks et al., 14 Apr 2025).
• The quasi-Hamiltonian double $\mathrm{D}(G)$ leads to quasi-symplectic structures on the group side.
• Over regular loci, one obtains open sets where Poisson or symplectic ranks are maximal, and slices (e.g., Kostant slices) provide transverse sections leading to global quotient structures and modular functor behaviors in TQFT (Crooks et al., 14 Apr 2025).

In the context of affine symplectic singularities $X$, a crepant partial resolution $\rho: X' \rightarrow X$ is covered by some $\mathbb{Q}$-factorial terminalization $\pi: Y \rightarrow X$. The geometry of $X'$ and its deformation theory is governed by the Poisson structure, with the deformation functor being pro-representable and unobstructed. The Namikawa Weyl group $W(X')$ for $X'$ is defined as a parabolic subgroup of $W(X)$ and reflects the birational geometry of the resolution (Malaney, 2023).

3. Representation-Theoretic and Sheaf-Theoretic Aspects

Partial Springer resolutions underlie the construction of partial Springer sheaves and related actions of relative Weyl groups:

• The perverse sheaf $$S_P := (\pi_P)_! k_{\widetilde{\mathcal{N}}_P}[d_P]$$ realizes Borel–Moore homology of fibers over the nilpotent cone and encodes intersection cohomology of nilpotent orbits (Chatterjee et al., 2024).
• There are two canonical $W(L)$-actions on $S_P$: the geometric (restriction) action and the Fourier-transform action. These differ by the character $\epsilon_P$ corresponding to the action of $W(L)$ on the top cohomology of $G/P$:

$\varphi_P = \rho_P \circ \Lambda_P$

where $\Lambda_P(w)=\epsilon_P(w)w$ (Chatterjee et al., 2024).

• The stalk cohomology of $S_P$ at $x$ is the Borel–Moore homology $H_*(\pi_P^{-1}(x))$, and over regular orbits, the irreducible components are permuted simply transitively by $W(L)$, mirroring classical Springer theory (Chatterjee et al., 2024).

In the setting of symmetric spaces, partial Springer resolutions are constructed over quasi-split symmetric pairs $(G, G_0)$ to resolve singularities over regular loci and realize actions of little Weyl groups $W_a$ on the cohomology of fibers, enabling a "partial Springer correspondence" for symmetric spaces (Leslie, 2019).

4. Modular, Combinatorial, and Enumerative Aspects

Partial Springer resolutions reveal deep combinatorial structures. For a reductive group $G$ and parabolic $P_J$ (Levi $L$), two types appear over $\mathbb{F}_q$:

• $\Spr_J^+=\{(u,yP_J) : u \in y V_J y^{-1}\}$, and
• $\Spr_J^-=\{(u,yP_J) : u \in y U_J y^{-1}\}$ where $V_J$ and $U_J$ denote varieties of unipotent elements and unipotent radicals (Trinh et al., 24 Jan 2026).

Central elements in the Iwahori–Hecke algebra $H_W$ are constructed via the Harish–Chandra transform and identified with relative-norm maps, providing new bases for the center of $H_W$ and enabling detailed point counts on braid Steinberg varieties and partial Springer fibers (Trinh et al., 24 Jan 2026). Deodhar cell decompositions in these spaces yield explicit enumerative results; associated noncrossing sets interpolate between rational Catalan, parking, and Kirkman numbers, generalizing (and interpolating) many classical counting results in Coxeter and braid groups.

The cohomology of generalized and partial Springer fibers admits a combinatorial description in terms of battery-powered tableaux, cocharge statistics, and skew Hall–Littlewood expansions. The Borho–MacPherson theory matches the isotypic components in Springer cohomology with skew operations on symmetric functions, enabling explicit Schur and charge formulas relevant to the Delta Conjecture (Gillespie et al., 2023).

5. Poisson Deformations and Weyl Group Symmetries

Partial resolutions in the symplectic and Poisson category exhibit sophisticated deformation theory:

• The Poisson deformation functor of a partial resolution $X'$ is prorepresentable by a formal power series ring and unobstructed if $H^i(X',\mathcal{O}_{X'})=0$ for $i>0$ (Malaney, 2023).
• There exists a Galois covering on formal deformation spaces with Galois group given by the Namikawa Weyl group $W(X')$, a parabolic subgroup of $W(X)$ determined by the associated face of the movable cone $\operatorname{Mov}(Y/X)$ (Malaney, 2023).
• The symplectic (or Springer) sheaf over the universal formal deformation of $X'$ realizes the regular $W(X')$-representation over the regular locus, and the endomorphism algebra of the Springer sheaf is naturally identified with the group algebra $\mathbb{C}[W(X')]$. Under rational smoothness hypotheses, one obtains the canonical Springer isomorphism

$$H^\ast(\rho^{-1}(0),\mathbb{Q}) \cong H^\ast(\pi^{-1}(0),\mathbb{Q})^{W(X')}$$

(Malaney, 2023).

6. Explicit Constructions: Type A and Toric Techniques

In type $A$, the partial Springer resolution for $P\subset GL_n$ (block upper-triangular matrices) is explicitly described: the zero fiber of the moment map

$$\mu: T^*(\mathfrak{p}\times \mathbb{C}^n)\to \mathfrak{p}^*,\quad \mu(r,s,i,j) = [r,s] + ij$$

gives a complete intersection of expected dimension when $P$ has finitely many orbits on its nilradical (Im et al., 2018). The GIT quotients of the zero fiber parametrize moduli of semistable quadruples, and for $P$ Borel, one recovers the classical case.

Further, toric constructions generalize Springer resolutions to universal covers of regular nilpotent orbits and to partial resolutions corresponding to intermediate toric subdivisions/lattices. These techniques provide birational, often locally orbifold, resolutions over covers $M$ of the nilpotent variety $N$, and yield explicit coordinate ring decompositions in terms of induced representations (Graham, 2019).

Construction	Base Variety	Fiber Description
$\widetilde{\mathcal{N}}_P$	Partial flag $G/P$	Variety of $P$-stable flags
$(T^*G)_\mathcal{C}$	Incidence $\mathfrak{g}_\mathcal{C}$	Integrates Poisson on $\mathfrak{g}_\mathcal{C}$
Toric $\widetilde M$	Universal nilpotent cover	Toric fiber structure; singular/regular fibers

7. Applications and Relationships to Other Theories

Partial Springer resolutions are essential in several advanced contexts:

• Topological Quantum Field Theories: Via restrictions to regular loci and Kostant/Steinberg slices, partial resolutions furnish symmetric monoidal functors from $2$-cobordisms to shifted symplectic categories. Canonical Lagrangian correspondences relate Hamiltonian symplectic varieties from partial TQFTs to open Moore–Tachikawa varieties (Crooks et al., 14 Apr 2025).
• Springer Theory for Symmetric Spaces: For quasi-split symmetric pairs, partial Springer resolutions yield simultaneous resolutions over the regular locus and enable “Springer correspondences” for little Weyl groups (Leslie, 2019).
• Hecke Algebra and Knot Invariants: Central elements arising from partial resolutions map, via the Harish–Chandra transform, to basis elements for the centers of Hecke algebras, with direct connections to HOMFLYPT polynomials and rational Kirkman numbers (Trinh et al., 24 Jan 2026).
• Generalized Springer Correspondence: The geometry and cohomology of partial Springer fibers underpin combinatorial and module-theoretic phenomena such as the structure of $\Delta$-Springer modules and provide geometric underpinnings for conjectures in algebraic combinatorics (Gillespie et al., 2023).

Taken together, partial Springer resolutions serve as a unifying geometric and categorical framework linking singularity theory, symplectic geometry, representation theory, and algebraic combinatorics, providing explicit bridges between geometry, topology, and enumerative structures across these disciplines.