Glauberman Correspondents in Finite Groups

• Glauberman correspondence is a canonical bijection that maps P-invariant irreducible characters of finite groups to those of the corresponding fixed-point subgroups.
• It establishes a degree-divisibility property where the degree of the fixed-point character divides that of the original, underpinning reductions in local-global conjectures.
• Advanced generalizations extend the correspondence to modular settings and block theory, enabling Morita equivalences and linking to moonshine phenomena in sporadic groups.

The Glauberman correspondence is a canonical bijection in the character theory of finite groups that links certain invariant irreducible characters under the action of a group of automorphisms to irreducible characters of a fixed-point subgroup. Originating from George Glauberman's work in the 1960s, this correspondence underpins structural results in representation theory, modular representation theory, block theory, and the formulation and reduction of local-global conjectures such as the Alperin Weight and McKay conjectures. Modern developments generalize the correspondence to more refined block-theoretic settings, facilitate Morita equivalences between certain block algebras, and link to moonshine phenomena in relation to the Monster group and lattice theory.

1. Definition and Construction of the Glauberman Correspondence

Let $G$ be a finite group and $p$ a prime. Given a $p$-subgroup $P$ of $\operatorname{Aut}(G)$ with $(|P|,|G|)=1$, the standard Glauberman correspondence establishes a canonical bijection

$\chi \mapsto \chi^*$

between the set of $P$-invariant irreducible complex characters of $G$, denoted $\operatorname{Irr}^P(G)$, and the set of irreducible characters of the fixed-point subgroup $$G^P = C_G(P) = \{g \in G \mid x(g)=g\ \forall x\in P\}$$. The character $\chi^*$ is uniquely determined as the constituent of the restriction $\operatorname{Res}^{G}_{G^P}(\chi)$ of maximal $p'$-degree: it is the unique irreducible constituent whose degree is not divisible by $p$, generalizing to the case where $P$ is solvable and acts coprimely on $G$ (Geck, 2019).

The construction extends to the context of blocks via the action of a $p'$-automorphism $\sigma$ of $G$. Watanabe's theorem associates to a $\sigma$-stable block $b$ of $\mathcal OH$ a unique block $b^\sigma$ of $\mathcal O(H^\sigma)$ (the Glauberman correspondent of $b$), under appropriate centralizing conditions on defect groups (Puig et al., 2012).

2. Fundamental Properties and Degree-Divisibility

A central property of the Glauberman correspondence is the divisibility relation between the degrees of corresponding characters. For $P$ a solvable $p$-subgroup of $\operatorname{Aut}(G)$ with $(|P|,|G|)=1$ and $\chi\in\operatorname{Irr}^P(G)$, Hartley and Turull proved that

$\chi^*(1)\ |\ \chi(1)$

where $\chi^*$ is the Glauberman correspondent of $\chi$ (Geck, 2019). The proof of this “degree-divisibility” property is based on congruence relations for Deligne–Lusztig Green functions associated with finite groups of Lie type. The key congruence (for a connected reductive group $G$ over $\mathbb{F}_q$, Frobenius map $F$, $F$-stable torus $T$ and prime $r\nmid |T^F|$) is: $Q_{T,F}(u)\ \equiv\ Q_{T,F^r}(u) \pmod r$ This removes prior restrictions on small characteristics and yields unconditional degree-divisibility for all finite groups. The ratio $\chi(1)/\chi^*(1)$ is always a product of primes dividing $|P|$; it is a $p$-power when $P$ is a $p$-group (Geck, 2019).

3. Block Theoretic Extensions and Morita Equivalence

The modular (block) theoretic Glauberman correspondence relates not only characters but also blocks of group algebras. If $G$ is a finite group and $\sigma$ is a solvable $p'$-automorphism, and $b$ is a $\sigma$-stable nilpotent block of $\mathcal O G$ with defect group $P$, under mild group-theoretic conditions there is a Morita equivalence

$$\mathcal O G\,b \simeq \mathcal O G^\sigma\,b^\sigma$$

where $b^\sigma$ is the Glauberman correspondent block in $\mathcal O(G^\sigma)$. The extension theory is governed by the structure of certain categories and interior algebras, and the Morita equivalence descends from the comparison of graded-internal structures of the original and correspondent blocks. This provides a unified conceptual framework for earlier results on Morita equivalence of blocks with normal defect groups and blocks covering nilpotent blocks (Puig et al., 2012).

4. Generalizations: Modular Characters, Brauer Theory, and Dade–Glauberman–Nagao Correspondence

The Dade–Glauberman–Nagao (DGN) correspondence generalizes the original correspondence to modular settings and defect-zero blocks. If $N\triangleleft M$ and $M/N$ is a $p$-group, and $\theta\in \operatorname{Irr}(N)$ is $M$-invariant of $p$-defect zero, then the DGN-correspondent $\theta^*$ is the unique defect-zero character of $C_N(D)$ for a defect group $D$ of the unique block covering $\theta$. Bijections exist at the level of irreducible (modular) characters and Brauer characters, functorial in conjugation and compatible with Galois automorphisms and automorphism groups.

Recent advances have extended the correspondence to Brauer characters and shown they preserve vertices (in the sense of representation theory), with full compatibility properties needed in reduction theorems for local-global conjectures (including Galois versions of the Alperin Weight Conjecture) (Fu, 2023). The technical machinery involves modular character triples, $\widehat H$-triples, and cohomological techniques ensuring blockwise–isomorphism bijections under precise functorial and equivariance constraints.

5. Role in Counting Conjectures and Equivariant Reductions

The Glauberman correspondence is crucial for reductions in local-global conjectures such as the Alperin Weight, McKay, and their Galois or automorphism-refined analogues. Navarro's "unified conjecture" merges the Alperin Weight Conjecture, Dade–Glauberman–Nagao correspondence, and modular equivalence of certain character triples into a single formalism, which has been recently shown to reduce to quasi-simple group cases under inductive conditions (Martínez et al., 2023). In these settings, the Glauberman correspondence supplies the bijection between $A$-invariant irreducible (ordinary or Brauer) characters of a group $G$ and the irreducible characters of a suitable fixed-point subgroup or centralizer, compatible with block structure and automorphism groups.

In block theory, this correspondence facilitates the formation of automorphism-equivariant bijections between Brauer characters and Alperin weights, each packaged as block isomorphisms of modular character triples. The full suite of compatibilities provided by the Glauberman and DGN correspondences—Galois, automorphism, restriction/induction, block and weight compatibility, and vertex preservation—are essential for reduction theorems that prove or reduce these conjectures for all finite groups once they are established for quasi-simple groups (Martínez et al., 2023, Fu, 2023).

6. Exotic Realizations and Connections with Moonshine

In a different direction, the Glauberman correspondence informs the "Glauberman–Norton" perspective in the context of moonshine and the Monster group. This setting connects dihedral automorphism types of involution pairs in the Monster $M$ to nodes in the extended $E_8$-Dynkin diagram, with central quotients of normalizers (centralizers of $2A$, $2B$ involutions and their products) conjecturally corresponding to “half Weyl groups” attached to subdiagrams with specific nodes deleted.

In concrete terms, orbits of triples $(x,y,z)$ (with $x,y$ in $2A$ and $z$ in $2B$ involutions, and $xy\in 3A$ or $6A$ Monster classes) correspond to configurations of $EE_8$-sublattices in the Leech lattice. The mapping to "half Weyl groups" is explicit: for the $3A$-node case, the central quotient is $(3\times PSU(4,2)).2$, half the size of $W(A_2)\times W(E_6)$, the Weyl group of the complementary $A_2+E_6$ subdiagram. These connections link certain properties of the Glauberman correspondence to vertex operator algebra theory and the structure of sporadic finite simple groups (Jr. et al., 2012).

7. Conclusion and Implications

The Glauberman correspondence is a cornerstone of finite group representation and block theory, with deep combinatorial and cohomological structure. Its extension from ordinary characters to blocks, modular, and equivariant contexts reflects its fundamental role in unifying disparate strands in the theory of finite groups. Its compatibility properties underlie pivotal reduction theorems for local-global conjectures, and its presence in moonshine phenomena and lattice theory indicates persistent links between character theory and other algebraic and geometric structures. Recent results have established increasingly refined equivariant and blockwise versions, resolving long-standing divisibility and equivalence problems and greatly advancing the conceptual clarity and reach of character-theoretic correspondences (Geck, 2019, Puig et al., 2012, Martínez et al., 2023, Fu, 2023, Jr. et al., 2012).