Equivariant Slodowy Slices

• Equivariant Slodowy slices are affine subspaces in complex Lie algebras, defined via an sl₂-triple and equipped with rich group symmetries.
• They serve as local models for nilpotent orbit deformations and symplectic singularities through universal Poisson deformation theory.
• Explicit algebraic constructions in both classical and exceptional types underscore their impact on representation theory and singularity analysis.

Equivariant Slodowy slices are affine subspaces constructed within a complex simple Lie algebra, defined through an associated sl$_2$-triple and equipped with group symmetries that reflect both the Lie algebra structure and additional equivariant data, such as Poisson automorphisms or group actions. These slices are central to the paper of the geometry and deformation theory of nilpotent orbits, symplectic singularities, and their quantizations, generalizing the classical constructions of Brieskorn and Slodowy and providing local models for singularities with rich equivariant structure.

1. Slodowy Slices: Definition and Transversality

A Slodowy slice $S$ in a simple complex Lie algebra $\mathfrak{g}$ is constructed starting from a nilpotent element $x$ and an associated sl$_2$-triple $\{x,h,y\}$ satisfying $[h,x]=2x, \ [h,y]=-2y, \ [x,y]=h$. The slice is defined as

$S = x + \Ker(\mathrm{ad}\ y)$

which is an affine subspace transverse to the adjoint orbit $G \cdot x$. The intersection of $S$ with the nilpotent cone $N = y^{-1}(0)$, denoted $S_0 = S \cap N$, yields the central fiber, encoding local geometric properties of the singularities in representation theory and algebraic geometry. In ADE cases, $S_0$ recovers Kleinian or Du Val surface singularities, with the Dynkin diagram structure matching that of $\mathfrak{g}$.

The adjoint quotient map $$y : \mathfrak{g} \to \mathfrak{g}/G \cong \mathfrak{h}/W$$, restricted to $S$, plays a central role in the deformation theory of these singularities. Slodowy slices inherit a Poisson structure from the restriction of the Lie bracket, making them prime examples of transverse symplectic slices.

2. Universal Poisson Deformations and Equivariance

A Poisson deformation of a singularity $X$ (e.g., $S_0$) is a flat family where the Poisson structure deforms compatibly. The deformation is called universal if any other (local or formal) Poisson deformation is uniquely realized as its pullback. The paper demonstrates that, for most nilpotent elements, the map

$p_S : S \to \mathfrak{h}/W$

is the formally universal Poisson deformation of $S_0$, provided the restriction map on cohomology

$$r: H^2(\widetilde{N}, \mathbb{Q}) \to H^2(F_x,\mathbb{Q})$$

(from the Springer resolution $\widetilde{N}$ to the Springer fiber $F_x$) is an isomorphism. The base of the deformation, $\mathfrak{h}/W$, encodes the action of the Weyl group and thus the full symmetry of the equivariant structure.

When coupled with a group action—such as a torus or the adjoint group—the equivariant structure ensures universality of the deformation within the category of group-equivariant Poisson varieties. For simply-laced types ($A$, $D$, $E$), universality always holds for non-regular nilpotent elements; in non-simply-laced cases, certain exceptions (classified in the paper) arise.

3. Classification of Nilpotent Orbits Admitting Universal Equivariant Deformations

The main classification result identifies for which nilpotent orbits the restriction of the adjoint quotient to the Slodowy slice yields a universal (equivariant) Poisson deformation. For non-regular nilpotent $x\in N$, universality typically holds except for specific orbits, including:

• Type $B_n$: the subregular orbit,
• Type $C_n$: orbits with Jordan types $[n,n]$ and $[2n - 2i, 2i]$ for $1 < i$,
• $G_2$: orbits of dimensions 8 and 10,
• $F_4$: the subregular orbit.

In simply-laced root systems, universality is robust under all non-regular nilpotents. This extends classical results (e.g., Brieskorn-Slodowy for subregular orbits) into the equivariant context and to higher-dimensional singularities, as shown in the paper's main theorems.

4. Singular Symplectic Hypersurfaces and Explicit Models

Equivariant Slodowy slices often produce new examples of singular symplectic hypersurfaces. Notably:

• In type $C_n$ ($\mathfrak{g} = \mathfrak{sp}_{2n}$), four-dimensional hypersurfaces of the form:

$f = a^2x + 2ab\,y + b^2z + (xz - y^2)^n$

in $\mathbb{C}[a,b,x,y,z]$.

• For the exceptional Lie algebra $G_2$, six-dimensional hypersurfaces generated by variables $a, b, c, p, q, r, s$ (degrees 2 and 3), with relations such as

$t_1 = a(ac-b^2) + 2(q^2 - rp)$

and further equations, culminating in

$$f = z^{\alpha} a - 2 z^{\alpha^2} b + z^{\alpha} c + 2(t_2-t_1t_3)$$

(after suitable rescaling).

The geometric interpretation is that these hypersurfaces provide local models that generalize Kleinian surface singularities, with Poisson and symplectic structures computed via resolutions of the Jacobian ideal of $f$. Their explicit algebraic descriptions enable further analysis and applications in symplectic singularity theory.

5. Mathematical Framework and Technical Formulations

Central formulas and constructions include:

• The definition of the Slodowy slice, $S = x + \Ker(\mathrm{ad}\ y)$;
• The adjoint quotient map $y : \mathfrak{g} \to \mathfrak{h}/W$ and its restriction $p_S$;
• The Poisson bracket induced from the Lie algebra:

$$\{-, -\} : \operatorname{Sym}^2(\mathfrak{g}) \to \mathfrak{g}$$

restricted to $S$.

• The tangent space to the Poisson deformation functor, interpretable as $H^2(S_0, \mathbb{C})$ through the Lichnerowicz–Poisson complex and truncated de Rham theory.

Explicit polynomial equations delineate new symplectic hypersurfaces, both in four and six dimensions. The technical results link these constructions to universal Poisson deformations via cohomological criteria on the Springer maps.

6. Equivariant Geometry, Quantum Cohomology, and Broader Implications

The universality of equivariant Slodowy slices implies their canonical role in controlling not just local geometry, but full deformation families under group actions. When the cohomological restriction is an isomorphism (as in simply-laced cases), the slice categorically parametrizes all equivariant Poisson deformations of its central fiber. This underlies applications such as:

• The explicit description of quantum multiplication operators in the quantum cohomology of Springer fibers.
• The construction of symplectic hypersurfaces with rigid Poisson structure, suitable as local models for more complex singularities.
• Compatibility with dual pairs formalism in Poisson geometry, connecting Slodowy slices to other Lie-theoretic symplectic varieties.

Equivariant universal deformations obtained through Slodowy slices also interact with Poisson deformations in the context of dual pairs, per Weinstein's theory, as elaborated in the paper.

7. Connections to Representation Theory and the Theory of Symplectic Singularities

Equivariant Slodowy slices furnish geometric models for various phenomena in representation theory, particularly:

• The categorification of quantum group actions and the structure theory of primitive ideals via finite W-algebras.
• The paper of symplectic singularities and their deformations, which underpin advances in geometric representation theory, moduli spaces, and algebraic geometry.
• The extension and generalization of classical results to new singular symplectic hypersurfaces, algebraic families with nontrivial group symmetry, and higher-dimensional models beyond surface singularities.

These approaches open new research avenues concerning local moduli spaces, quantizations, and birational geometry of symplectic singularities with equivariant structure.

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In summary, equivariant Slodowy slices emerge as fundamental objects integrating Lie algebraic transversality, Poisson and symplectic deformation theory, and equivariant geometric structure. Their universality properties—especially in simply-laced types—enable broad applications in singularity theory, quantum cohomology, and representation theory, and their explicit algebraic models inform ongoing developments in the classification and analysis of symplectic singularities and their moduli.