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McKay's conjecture (1971) on character degrees was reduced by Isaacs-Malle-Navarro (2007) to a so-called inductive condition on characters of finite quasisimple groups [IMN07], thus opening the way to a proof of McKay's conjecture using the classification of finite simple groups. After [Ma07], [Ma08], [S12], [CS13], [KS16], [MS16], [CS17a], [CS17b], [CS19], [S23a], [S23b], we complete here the last step of a proof with an analysis of the representations of certain normalizers ${\mathrm N}_G({\mathbf S})$ in $G={\mathbf G}^F$ of maximal $d$-tori ${\mathbf S}$ ($d\geq 3$) of the ambient simple simply-connected algebraic group ${\mathbf G}$ of type ${\mathrm D}_l$ ($l\geq 4$) for which $F$ is a Frobenius endomorphism. To establish the so-called local conditions A$(d)$ and B$(d)$, we introduce a certain class of $F$-stable reductive subgroups ${\mathbf M}\leq {\mathbf G}$ of maximal rank where ${\mathbf M}^\circ$ is of type ${\mathrm D}_{l_1}\times {\mathrm D}_{l-l_1}$ with ${\mathbf M}/{\mathbf M}^\circ$ of order 2. They are an efficient substitute for ${\mathrm N}_G({\mathbf S})$ or the local subgroups in non-defining characteristic relevant to McKay's abstract statement. For a general class of those subgroups ${\mathbf M}^F$ we describe their characters and the action of $\operatorname {Aut}({\mathbf G}^F)_{{\mathbf M}^F}$ on them, showing in particular that $\mathrm{Irr}({\mathbf M}^F)$ and $\mathrm {Irr}({\mathbf G}^F)$ share some key features in that regard. With this established, McKay's conjecture is now a theorem stating $\textit{McKay's equality}$: For any prime $\ell$, any finite group has as many irreducible complex characters of degree prime to $\ell$ as the normalizers of its Sylow $\ell$-subgroups.

We determine the action of the outer automorphism group Out$(G)$ on the set of irreducible characters Irr$(G)$ for all finite quasi-simple groups $G$. For groups of Lie type, this includes the construction of an Out$(G)$-equivariant Jordan decomposition of characters (Theorem B). We also prove an extendibility statement, called condition $A(\infty)$, see Theorem A. Among the methods used here in the case of groups of type D and $^2$D, we blend the Shintani descent ideas introduced in [CS19, \S~3] and an analysis of semisimple classes in the dual group $G^*$ from [CS22]. This enables us to control the action of graph and field automorphisms within Lusztig's rational series of characters and to count extendible characters. Condition $A(\infty)$ originates in the program to prove the McKay conjecture using the classification of finite simple groups, see [IMN07]. Using the local results from [MS16], our main theorem implies the McKay conjecture for the prime 3 and all finite groups.

In adjoint reductive groups $H$ of type $\mathsf{D}$ we show that for every semisimple element $s$, its centralizer splits over its connected component, i.e., $C_H(s) = C_H(s)^\circ \rtimes \check A$ for some complement $\check A$ with strong stability properties. We derive several consequences about the action of automorphisms on semisimple conjugacy classes. This helps to parametrize characters of the finite groups $\mathsf{D}_{l,\text{sc}}(q)$ and $^2\mathsf{D}_{l,\text{sc}}(q)$ and describe the action of automorphisms on them. It is also a contribution to the final proof of the McKay conjecture for the prime 3, see [S21], [S23]., Comment: 11 pages, to appear in Vietnam J. Math

We show that the decomposition matrix of a given group $G$ is unitriangular, whenever $G$ has a normal subgroup $N$ such that the decomposition matrix of $N$ is unitriangular, $G/N$ is abelian and certain characters of $N$ extend to their stabilizer in $G$. Using the recent result by Brunat--Dudas--Taylor establishing that unipotent blocks have a unitriangular decomposition matrix, this allows us to prove that blocks of groups of quasi-simple groups of Lie type have a unitriangular decomposition matrix, whenever they are related via Bonnaf\'e--Dat--Rouquier's equivalence to a unipotent block. This is then applied to study the action of automorphisms on Brauer characters of finite quasi-simple groups. We use it to verify the so-called {\it inductive Brauer--Glaubermann condition}, that aims to establish a Glauberman correspondence for Brauer characters, given a coprime action.

This is a contribution to the study of $\operatorname {Irr}(G)$ as an $\operatorname {Aut}(G)$-set for $G$ a finite quasi-simple group. Focusing on the last open case of groups of Lie type $\mathrm D$ and $^2\mathrm D$, a crucial property is the so-called condition $A'(\infty)$ expressing that diagonal automorphisms and graph-field automorphisms of $G$ have transversal orbits in $\operatorname {Irr}(G)$. This is part of the stronger $A(\infty)$ condition introduced in the context of the reduction of the McKay conjecture to a question on quasi-simple groups. Our main theorem is that a minimal counter-example to condition $A(\infty)$ for groups of type $\mathrm D$ would still satisfy $A'(\infty)$. This will be used in a second paper to fully establish $A(\infty)$ for any type and rank. The present paper uses Harish-Chandra induction as a parametrization tool. We give a new, more effective proof of the theorem of Geck and Lusztig ensuring that cuspidal characters of arbitrary standard Levi subgroups of $G={\mathrm D}_{ l,\mathrm{sc}}(q)$ extend to their stabilizers in the normalizer of that Levi subgroup. This allows to control the action of automorphisms on these extensions. From there Harish Chandra theory leads naturally to a detailed study of associated relative Weyl groups and other extendibility problems in that context., Comment: 55 pages, to appear in Nagoya Mathematical Journal

We give a criterion that simplifies the checking of the inductive Alperin weight condition for the remaining open cases of simple groups of Lie type. It is strongly related in form to the criterion of the second author for the inductive McKay conditions (see [Sp\"a12,2.12]) that has proved very useful. The proof follows from a Clifford theory for weights intrinsically present in the proof of reduction theorems of the Alperin weight conjecture given by Navarro--Tiep and the second author. We also give a related criterion for the inductive blockwise Alperin weight condition.

The Alperin-McKay conjecture relates height zero characters of an $\ell$-block with the ones of its Brauer correspondent. This conjecture has been reduced to the so-called inductive Alperin-McKay conditions about quasi-simple groups by the third author. Those conditions are still open for groups of Lie type. The present paper describes characters of height zero in $\ell$-blocks of groups of Lie type over a field with $q$ elements when $\ell$ divides $q-1$. We also give information about $\ell$-blocks and Brauer correspondents. As an application we show that quasi-simple groups of type $C$ over $\mathbb{F}_q$ satisfy the inductive Alperin-McKay conditions for primes $\ell\geq 5$ and dividing $q-1$. Some methods to that end are adapted from the work of Malle--Sp\"ath.

We generalize the theory of ordering character triples, developed by Navarro and Sp\"ath, by taking into account the action of Galois automorphisms on characters. This new technique, together with previous results of Ladisch and Turull, allows us to reduce the Galois--McKay conjecture to a question about simple groups., Comment: 30 pages

We establish the inductive McKay condition introduced by Isaacs-Malle-Navarro \cite{IMN} for finite simple groups of Lie types $\tB_l$ ($l\geq 2$), $\tE_6$, $^2\tE_6$ and $\tE_7$, thus leaving open only the types $\tD$ and $^2\tD$. We bring to the methods previously used by the authors for type $\tC$ \cite{CS17C} some descent arguments using Shintani's norm map. This provides for types different from $ \tA, \tD, {}^2\tD$ a uniform proof of the so-called global requirement of the criterion given by the second author in \cite[2.12]{S12}. The local requirements from that criterion are verified through a detailed study of the normalizers of relevant Levi subgroups and their characters., Comment: 41 pages

In this paper characters of the normaliser of $d$-split Levi subgroups in $\mathrm {SL}_n(q)$ and $\mathrm {SU}_n(q)$ are parametrized with a particular focus on the Clifford theory between the Levi subgroup and its normalizer.These results are applied to verify the Alperin-McKay conjecture for primes $\ell$ with $\ell\nmid 6(q^2-1)$ and the Alperin weight conjecture for $\ell$-blocks of those quasi-simple groups with abelian defect. The inductive Alperin-McKay condition and inductive Alperin weight condition by the second author are verified for certain blocks of $\mathrm {SL}_n(q)$ and $\mathrm {SU}_n(q)$.

We verify the inductive McKay condition for simple groups of Lie type C, namely finite projective symplectic groups. This contributes to the program of a complete proof of the McKay conjecture for all finite groups via the reduction theorem of Isaacs-Malle-Navarro and the classification of finite simple groups. In an important step we use a new counting argument to determine the stabilizers of irreducible characters of a finite symplectic group in its outer automorphism group. This is completed by analogous results on characters of normalizers of Sylow d-tori in those groups.

Suppose that $B$ is a Brauer $p$-block of a finite group $G$ with a unique modular character $\varphi$. We prove that $\varphi$ is liftable to an ordinary character of $G$ (which moreover is $p$-rational for odd $p$). This confirms the basic set conjecture for these blocks.

We prove the McKay conjecture on characters of odd degree. A major step in the proof is the verification of the inductive McKay condition for groups of Lie type and primes $\ell$ such that a Sylow $\ell$-subgroup or its maximal normal abelian subgroup is contained in a maximally split torus by means of a new equivariant version of Harish-Chandra induction. Specifics of characters of odd degree, namely that they only lie in very particular Harish-Chandra series then allow us to deduce from it the McKay conjecture for the prime~$2$, hence for characters of odd degree.

If a finite group $A$ acts coprimely as automorphisms on a finite group $G$, then the $A$-invariant Brauer $p$-blocks of $G$ are exactly those that contain $A$-invariant irreducible characters.

As a sequel to [CS13b], we verify the so-called inductive AM-condition introduced in [Sp12] for simple groups of type A and blocks with maximal defect. This is part of the program set up to verify the Alperin-McKay conjecture through its reduction to a problem on quasi-simple groups (see [Sp13]) but also the missing direction of Brauer's height zero conjecture (see [NS14]), Comment: 16 pages

We verify the inductive blockwise Alperin weight (BAW) and the inductive Alperin-McKay (AM) conditions introduced by the second author for blocks of finite quasisimple groups with cyclic defect groups. Furthermore we establish a criterion that describes conditions under which the inductive AM condition for blocks with abelian defect groups implies the inductive BAW condition for those blocks.

We present a new criterion to predict if a character of a finite group extends. Let $G$ be a finite group and $p$ a prime. For $N\lhd G$, we consider $p$-blocks $b$ and $b'$ of $N$ and ${\rm N}_N(D)$, respectively, with $(b')^N=b$, where $D$ is a defect group of $b'$. Under the assumption that $G$ coincides with a normal subgroup $G[b]$ of $G$, which was introduced by Dade early 1970's, we give a character correspondence between the sets of all irreducible constituents of $\phi^G$ and those of $(\phi')^{{\rm N}_G(D)}$ where $\phi$ and $\phi'$ are irreducible Brauer characters in $b$ and $b'$, respectively. This implies a sort of generalization of the theorem of Harris-Kn\"orr. An important tool is the existence of certain extensions that also helps in checking the inductive Alperin-McKay and inductive Blockwise Alperin Weight conditions, due to the second author.

As a step to establish the McKay conjecture on character degrees of finite groups, we verify the inductive McKay condition introduced by Isaacs-Malle-Navarro for simple groups of Lie type $A_{n-1}$, split or twisted. Key to the proofs is the study of certain characters of SL$_n(q)$ and SU$_n(q)$ related to generalized Gelfand-Graev representations. As a by-product we can show that a Jordan decomposition for the characters of the latter groups is equivariant under outer automorphisms. Many ideas seem applicable to other Lie types.

We reformulate the inductive McKay condition, from Isaacs-Malle-Navarro, and apply the new criterion to simple groups of Lie type, when the prime is the defining characteristic p. Thereby we make use of a recent result of Maslowski. This proves that these simple group satisfy the inductive McKay condition for p. In the non simply-laced types and non-classical types this reproves earlier results by Brunat and Brunat-Himstedt., Comment: 12 pages

We prove for finite reductive groups $G$ of classical type, that every irreducible character of $L$ extends to its inertia group in $N$, where $L$ is an abelian centraliser of a Sylow $d$-torus $\mathbf S$ of $G$ and $N:=N_G(\mathbf S)$. This gives a precise description of the irreducible characters of $N$. Furthermore it enables us to verify the McKay conjecture in this situation for $G$ and some primes.