Topologically Nilpotent Regular Semisimple Element

• Topologically nilpotent regular semisimple elements are defined in Lie algebras of reductive groups over discretely valued fields, characterized by minimal centralizers, diagonalizable adjoint actions, and eigenvalue constraints from the valuation topology.
• They play a critical role in the representation theory of p-adic groups and the study of affine Springer fibers, providing canonical representatives via constructions like the Kostant slice.
• Explicit algorithms incorporating Newton polygons and partition rules in classical types establish their minimal reduction types, thereby clarifying stable conjugacy classes and aiding computational approaches.

A topologically nilpotent regular semisimple element is a distinguished class of elements in the Lie algebra of a reductive group over a field complete with respect to a discrete valuation. Such an element is simultaneously regular (its centralizer is as small as possible), semisimple (its adjoint action is diagonalizable), and topologically nilpotent (its "size", in terms of eigenvalues or adjoint quotient invariants, is constrained by the valuation topology in a manner that links to reduction modulo the maximal ideal). These elements play a central role in the representation theory of $p$-adic groups, the theory of affine Springer fibers, and the construction of canonical sections such as the Kostant slice, offering unique representatives for stable conjugacy classes under strong structural constraints (Adler et al., 2016, Wang et al., 11 Jan 2026).

1. Definitions and Foundational Properties

Let $F$ be a field complete with respect to a discrete valuation $v:F^{\times}\to\mathbf{Z}$, with ring of integers $\mathcal{O}_F$, maximal ideal $\mathfrak{p}_F$, and perfect residue field $k_F$. Let $G$ be a connected, quasi-split reductive group over $F$, and $\mathfrak{g}$ its Lie algebra.

Given $X\in\mathfrak{g}(F)$ with Jordan decomposition $X=X_s+X_n$ (with $X_s$ semisimple, $X_n$ nilpotent), $X$ is called topologically nilpotent if, for every algebraic character $\alpha$ of an $F$-torus $T\subset G$ (with $\operatorname{Lie}(T)$ containing $X_s$), $|d\alpha(X_s)|<1$. This characterization is independent of the choice of $T$, and equivalently, $X$ is topologically nilpotent precisely if its image under the adjoint quotient map $\chi:\mathfrak{g}\to\mathfrak{q}:=\mathfrak{g}//G$ lies in the reduction-mod-$\mathfrak{p}_F$ fiber $\mathfrak{q}(\mathcal{O}_F)\to\mathfrak{q}(k_F)$ over the origin (Adler et al., 2016).

An element is regular semisimple if its centralizer is a maximal torus (i.e., of minimal dimension equal to the rank of $G$), and the adjoint operator is diagonalizable over a separable closure.

2. Explicit Characterizations and Affine Settings

For loop groups $LG$ over $F=\mathbf{C}((t))$, with $\mathcal{O}=\mathbf{C}[[t]]$ and $L^+=\mathcal{O}$, the condition of topological nilpotence for $\gamma\in L\mathfrak{g}$ is witnessed by all eigenvalues (in a splitting field) lying in the maximal ideal of $\mathcal{O}$, equivalently, the constant term $a_n(t)$ in the characteristic polynomial $$\chi_\gamma(\lambda)=\lambda^n + a_1(t)\lambda^{n-1}+\cdots+a_n(t)$$ satisfies $\operatorname{ord}_t a_n > 0$ (Wang et al., 11 Jan 2026). Regular semisimplicity in this context again means the centralizer is a split maximal torus.

3. Stable Conjugacy and the Adjoint Quotient

Two regular semisimple elements $X,X'\in\mathfrak{g}(F)$ are stably conjugate if they are $G(\overline{F})$-conjugate and their orbits correspond under the Galois group. The fibers of the adjoint quotient map $\chi:\mathfrak{g}(F)\to\mathfrak{q}(F)$ parametrize such stable conjugacy classes among the regular semisimple locus. Topologically nilpotent regular semisimple elements correspond to the fiber over the origin in the reduction modulo $\mathfrak{p}_F$ (Adler et al., 2016).

4. Canonical Representatives: Kostant Section and Slices

Under mild conditions (quasi-split $G$, sufficient residue characteristic, certain tameness hypotheses), the construction of an integral Kostant section provides canonical representatives for each stable, regular, topologically nilpotent class. Fixing a regular nilpotent $Y$ determined by a pinning, and a hyperspecial point $x$ in the Bruhat–Tits building, the Kostant slice

$Y + \mathfrak{g}(F)_{x,0+} \subset \mathfrak{g}(F)$

intersects each $G(F)$-orbit in the set of stable, regular, topologically nilpotent conjugacy classes exactly once. In particular, for $G=GL_n$, this recovers the familiar "companion-matrix" slice structure (Adler et al., 2016).

5. Minimal Reduction Types: Affine Grassmannian and Newton Polygon Algorithms

Given a topologically nilpotent regular semisimple $\gamma$ in $L\mathfrak{g}$, the minimal reduction conjecture of Yun states that the corresponding set of minimal nilpotent orbits realized by reduction modulo $t$ in the affine Grassmannian is a singleton. This was established for all classical types, with the minimal reduction determined explicitly using the Newton polygon of the characteristic polynomial $\chi_\gamma$ (Wang et al., 11 Jan 2026).

The explicit procedure is as follows:

1. Factor $\chi_\gamma(\lambda)=\prod_i f_i(\lambda)$, with each $f_i$ of single slope (i.e., with $\operatorname{ord}_t f_i(0)$ constant).
2. For each $f_i$, express $\deg f_i=n_i=m_i k_i + l_i$, $0\le l_i<m_i$.
3. Form the A-type balanced partition $[(k_i+1)^{l_i},\; k_i^{m_i-l_i}]$.
4. In types C and D, or B, adjust as dictated by self-duality and parity constraints: apply combinatorial rules analogous to those of Spaltenstein for orthogonal or symplectic partitions to obtain the unique minimal admissible refinement.
5. The resulting partition describes the unique nilpotent orbit in the reduction.

This approach is effective and applies uniformly in all classical types, capturing intricate parity conditions in types B and D via explicit, algorithmic combinatorial corrections (Wang et al., 11 Jan 2026).

Group Type	Key Partition Rule	Additional Adjustments
Type A	A-type balanced partition	None
Type C	Self-dual A-type partition	Parity check for symplectic condition
Type D	A-type + parity/combinatorial rule	Admissibility via splitting/pairings
Type B	A-type+extra [1]+rule $(\star\star)$	Absorb/adjust last $[1]$-block as needed

6. Interaction with Affine Springer Fibers and Endoscopy

The theory extends to the study of affine Springer fibers, where, for a regular semisimple topologically nilpotent $\gamma$, the affine Springer fiber

$$\mathrm{Gr}_\gamma = \{gL^+G \mid g^{-1} \gamma g \in L^+\mathfrak{g}\}$$

admits a stratification by nilpotent orbits upon reduction, with the minimal reduction partition governing the geometry and representation-theoretic structure.

For $p$-adic fields, the characteristic function on the Kostant slice,

$\varphi_G = 1_{Y+\mathfrak{g}(F)_{x,0+}},$

has orbital integrals with sharp support on the corresponding single regular, topologically nilpotent class, and these distributions behave well under endoscopic transfer (the relative orbital integrals of endoscopic groups match after normalization) (Adler et al., 2016).

7. Examples and Canonical Forms

Concrete instances in each classical type illustrate the main algorithms:

• For $GL_n$, $\chi_\gamma(\lambda)=\lambda^3+t^2$ yields partition $[2,1]$.
• For $Sp_4$, $$\chi_\gamma(\lambda)=\lambda^4 + t^3\lambda^2 + t^5$$ with $m=5$, $n=4$, gives $[1^4]$ (regular nilpotent).
• For $SO_{10}$, a slope block of degree 10, $m=4$, $10=4\cdot2+2$, gives A-type $[3^2,2^2]$, which is already admissible.
• For $SO_7$, with characteristic polynomial $\lambda(\lambda^6 + t^3\lambda^4 + \cdots + t^5)$ and $m=5$, corresponding minimal reduction is $[3,1^4]$ after absorbing $[1]$ via the combinatorial rule (Wang et al., 11 Jan 2026).

These constructions provide explicit, canonical representatives in each stable, regular, topologically nilpotent class.

• B. Wang, X. Wen, Y. Wen. "Minimal reduction type in classical cases" (Wang et al., 11 Jan 2026).
• J.-K. Yu, J. Adler. "On Kostant Sections and Topological Nilpotence" (Adler et al., 2016).