Unipotent Similarity Classes

• Unipotent similarity classes are G-conjugacy classes of unipotent elements in reductive groups, defined via partitions for GLₙ and weighted Dynkin diagrams for exceptional types.
• They are parametrized using combinatorial data, including partitions with parity constraints in classical groups and additional invariants in bad characteristic cases.
• These classes underpin correspondences with Weyl group representations through the Springer map and inform closure relations essential for geometric representation theory.

A unipotent similarity class in the context of algebraic groups and their representations is a G-conjugacy class of a unipotent element in a reductive algebraic group G, or equivalently, under certain correspondences, an orbit of nilpotent elements in the Lie algebra Lie(G). The problem of classifying and analyzing these classes is central to the structure theory of algebraic groups, geometric representation theory, and the theory of group actions on varieties. The theory incorporates deep combinatorial, geometric, and representation-theoretic structures and is intimately connected with the Springer correspondence, the theory of Weyl groups, and various forms of canonical parametrizations.

1. Classification of Unipotent Similarity Classes

For a connected reductive group G over an algebraically closed field, unipotent similarity classes are G-orbits of unipotent elements, i.e., elements u satisfying $(u-1)^N = 0$ for some $N$. The classification of these classes depends crucially on the type of G and the characteristic of the ground field:

• Type A$_{n-1}$ (GL$_n$, SL$_n$): Classes correspond bijectively to partitions of $n$, labeled by the sizes of the Jordan blocks in the usual normal form. Every unipotent similarity class in GL$_n$ is determined by a partition $\lambda \vdash n$ (Lusztig, 2020, Korhonen et al., 2021).
• Types B$_n$, C$_n$, D$_n$ (SO/Sp groups): Classes are parametrized by special partitions of the group dimension, subject to parity constraints:
  • In type C$_n$ (Sp$_{2n}$), the partition must have all odd parts appearing with even multiplicity.
  • In type B$_n$ (SO$_{2n+1}$), even parts appear with even multiplicity.
  • In type D$_n$ (SO$_{2n}$), even parts appear with even multiplicity and, if all parts are even, each such partition gives rise to two distinct classes (Andruskiewitsch et al., 2014, 0912.3820).
  • In "bad" characteristic (notably $p=2$ for classical groups), finer invariants—such as the $\epsilon$-invariant of Spaltenstein—are required to fully classify classes (Lusztig et al., 2011, 0912.3820).
• Exceptional Types (G$_2$, F$_4$, E$_6$, E$_7$, E$_8$): Parametrization is via weighted Dynkin diagrams, with each admissible labelling of diagram nodes by 0, 1, or 2 corresponding to a unique unipotent class (Lusztig, 2020, Korhonen et al., 2021).

This structure explains the canonical finite partition of unipotent elements in each group and underpins the subsequent representation-theoretic correspondences.

2. Parametrization and Canonical Representatives

The canonical forms of unipotent similarity classes are typically described as follows:

• GL$_n$ Case: Each class corresponds to the conjugacy class of a Jordan matrix with block sizes as specified by the partition $\lambda$. Explicit representatives are given by the identity plus a canonical nilpotent matrix in symmetric form (Clarke, 2010).
• Classical Groups: For symplectic and orthogonal groups, representatives must preserve additional form constraints. Canonical representatives are constructed by embedding the class into GL$_n$, then adjusting to respect the alternating or quadratic forms, sometimes requiring sign modifications in certain matrix entries. The relevant combinatorics involve dual partitions, the multiplicities of odd and even parts, and for $p=2$ fields, additional invariants attached to certain block sizes (Lusztig et al., 2011, Korhonen et al., 2021, 0912.3820).
• Exceptional Groups: Chevalley basis techniques and explicit root group products (following a chosen ordering of roots and precise structure constants) yield canonical representatives for each class, as tabulated for the "eminent" classes in (Korhonen et al., 2021).

This canonical structure is reflected in various correspondence theorems (notably those of Springer and Bala–Carter), equipping each class with precise, computable invariants beyond mere Jordan block data.

3. Closure Relations and Partial Orders

The geometry of unipotent similarity classes admits a "closure order" (corresponding to Zariski closure of orbits), which can be described combinatorially:

• Type A: The closure ordering is given by the dominance order on partitions, i.e., $\lambda \leq \mu$ if and only if $\sum_{j=1}^{i}\lambda_j \leq \sum_{j=1}^{i}\mu_j$ for all $i$ (0912.3820).
• Types B, C, D: The closure relation on unipotent classes is described via product dominance orders on certain pairs of associated partitions, or, equivalently, on combinatorial invariants associated to the class (such as the ($c, \epsilon$) parametrization of Spaltenstein). More refined inequalities involving the partitions and extra invariants determine closure (Lusztig et al., 2011, 0912.3820).
• Correspondence with Bruhat Order: For Weyl group parametrizations, there is an equivalence between the closure order of unipotent classes and the Bruhat order of certain elliptic conjugacy classes, further underlining the geometric-combinatorial duality (Lusztig et al., 2011).

The closure order is of key importance in decomposition of the unipotent variety, the construction of "unipotent pieces", and the analysis of representation-theoretic induction and restriction functors.

4. Correspondence with Weyl Group Conjugacy and the Springer Map

One of the major results is the connection (or surjection) between Weyl group conjugacy classes and unipotent similarity classes. Lusztig defined a map $\Phi_p: \underline{W} \to$ {unipotent classes in $G$}, taking a conjugacy class in the Weyl group $W$ (or a twisted version for disconnected groups) to a unique unipotent class in G (Lusztig, 2011, Lusztig, 2011). Key features include:

• Surjectivity and Canonical Inverse: The map $\Phi_p$ is surjective, and there exists a canonical one-sided inverse mapping each unipotent class to a unique "special" conjugacy class in $W$ minimizing the kernel dimension of $w-1$ (Lusztig, 2011).
• Fibers and Similarity Classes: The fibers of $\Phi_p$ correspond to "unipotent similarity classes," or Olsson–Spaltenstein classes, with a direct combinatorial or representation-theoretic description (Lusztig, 2011).
• Special Conjugacy and Representations: The so-called special conjugacy classes in $W$ (images of Lusztig's canonical inverse) correspond bijectively with special representations of $W$, reinforcing the link between geometric partitions and representation-theoretic parameters (Lusztig, 2011).
• Bad Characteristic Modifications: For classical groups in "bad characteristic" (notably $p=2$), Lusztig and Xue proved that the surjection extends (for elliptic classes) and is characterized by a minimality and closure property (Lusztig et al., 2011). Specifically, for $C_n$ and $D_n$, elliptic Weyl group classes correspond via a strictly decreasing sequence to distinguished unipotent classes with specified ("all $1$") $\epsilon$-invariant, and closure relations follow the prescribed inequalities.

This framework enables a robust parametrization of unipotent similarity classes and explicit correspondence with Weyl-theoretic data for both connected and disconnected or twisted group settings (Lusztig, 2011).

5. Applications, Recognition, and Invariants

The structure of unipotent similarity classes has far-reaching implications:

• Centralizer and Representation Invariants: The structure and dimension of the centralizer in $G$ of a unipotent element is a key invariant. In exceptional types, further invariants (Jordan block sizes on minimal and adjoint modules, derived series dimensions, algebra of derivations, nilradicals, etc.) are sufficient to uniquely pin down classes (Korhonen et al., 2021).
• Springer Correspondence: Each unipotent class corresponds canonically to an irreducible representation of the Weyl group via the top-degree cohomology of the Springer fibre, providing a geometric realization of the set of irreducible $W$-modules (Lusztig, 2020).
• Unipotent Pieces: Lusztig, Mizuno, and Spaltenstein defined "unipotent pieces"—partitions of the unipotent variety into smooth, $G$-stable, locally closed subsets—coinciding precisely with the combinatorial fibers of the aforementioned maps, independent of field characteristic (0912.3820).
• Parametrizations in Non-split/Disconnected Groups: Explicit constructions and canonical forms for unipotent similarity classes extend to finite unitary, symplectic, and quaternionic groups via reduction and canonical isomorphisms, with clear formulas for orbit sizes and types (Dolžan, 23 Dec 2025, Andruskiewitsch et al., 2014).

Recognition theorems, combinatorial algorithms for canonical forms (Clarke, 2010, Tsai et al., 2020), and explicit representative tables all facilitate computational applications and further categorical/representation-theoretic analysis.

6. Examples and Special Cases

Tabulated descriptions and explicit examples illustrate the principles in small or special cases:

Group	Parametrization	Example Representative
GL$_n$	Partition $\lambda \vdash n$	$u_\lambda=I+N_\lambda$,
Sp$_{2n}$, $p\neq 2$	Symplectic partitions with even multiplicity on odd parts	Block-diag of $J_{\lambda_i}(1)$
SO$_n$, $p\neq 2$	Orthogonal partitions with parity constraints	As above, parity-checked
Type D$_n$, $p=2$	Partitions with Spaltenstein's $\epsilon$-invariants	$(c,\epsilon)$ (Lusztig et al., 2011)
Exceptional	Weighted Dynkin diagrams	Product over specified root elements (Korhonen et al., 2021)

For instance, in $Sp_4$ (type $C_2$), the four unipotent classes are in bijection with the partitions $(1^4), (2,1^2), (2^2), (4)$, with closure ordering by dominance and explicit representatives as prescribed in the symmetric canonical forms (0912.3820, Clarke, 2010). In $C_3$, the elliptic Weyl group classes give rise to unipotent classes with partitions $(3)$ (one block size 6) or $(2,1)$ (block sizes $4,2$), each with an "all $1$" $\epsilon$-invariant (Lusztig et al., 2011).

The full computational structures (orbit sizes, class counts, recognition invariants) are available for all families and are tabulated in the references cited above.

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References:

(Lusztig et al., 2011, Lusztig, 2011, Lusztig, 2011, Korhonen et al., 2021, 0912.3820, Lusztig, 2020, Clarke, 2010, Tsai et al., 2020, Andruskiewitsch et al., 2014, Dolžan, 23 Dec 2025)