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Let $G$ be a complex simply-laced semisimple algebraic group of rank $r$ and $B$ a Borel subgroup. Let $\mathbf i \in [r]^n$ be a word and let $\mathbf \ell = (\ell_1,\dots,\ell_n)$ be a sequence of non-negative integers. Grossberg and Karshon introduced a virtual lattice polytope associated to $\mathbf i$ and $\mathbf \ell$ called a twisted cube, whose lattice points encode the character of a $B$-representation. More precisely, lattice points in the twisted cube, counted with sign according to a certain density function, yields the character of the generalized Demazure module determined by $\mathbf i$ and $\mathbf \ell$. In recent work, the author and Harada described precisely when the Grossberg-Karshon twisted cube is untwisted, i.e., the twisted cube is a closed convex polytope, in the situation when the integer sequence $\mathbf \ell$ comes from a weight $\lambda$ of $G$. However, not every integer sequence $\mathbf \ell$ comes from a weight of $G$. In the present paper, we interpret untwistedness of Grossberg-Karshon twisted cubes associated to any word $\mathbf i$ and any integer sequence $\mathbf \ell$ using the combinatorics of $\mathbf i$ and $\mathbf \ell$. Indeed, we prove that the Grossberg-Karshon twisted cube is untwisted precisely when $\mathbf i$ is hesitant-jumping-$\mathbf \ell$-walk-avoiding., Comment: Keywords: Grossberg-Karshon twisted cubes, pattern avoidance, character formula, generalized Demazure modules. arXiv admin note: text overlap with arXiv:1407.8543

According to a 2018 preprint by Nobuaki Yagita, the conjecture on a relationship between [Formula: see text]- and Chow theories for a generically twisted flag variety of a split semisimple algebraic group [Formula: see text], due to the author, fails for [Formula: see text] the spinor group [Formula: see text]. Yagita’s tools include a Brown–Peterson version of algebraic cobordism, ordinary and connective Morava [Formula: see text]-theories, as well as Grothendieck motives related to various cohomology theories over fields of characteristic [Formula: see text]. We provide a proof using only the [Formula: see text]- and Chow theories themselves and extend the (slightly modified) example to arbitrary characteristic.

Let $G$ be a semisimple algebraic group over a field of characteristic $p > 0$. We prove that the dual Weyl modules for $G$ all have $p$-filtrations when $p$ is not too small. Moreover, we give applications of this theorem to $p^n$-filtrations for $n > 1$, to modules containing the Steinberg module as a tensor factor, and to the Donkin conjecture on modules having $p$-filtrations., minor revisions and updates

We introduce a rigid analytification of the quantized coordinate algebra of a semisimple algebraic group and a quantized Arens-Michael envelope of the enveloping algebra of the corresponding Lie algebra, working over a non-archimedean field and when $q$ is not a root of unity. We show that these analytic quantum groups are topological Hopf algebras and Fr\'echet-Stein algebras. We then introduce an analogue of the BGG category $\mathcal{O}$ for the quantum Arens-Michael envelope and show that it is equivalent to the category $\mathcal{O}$ for the corresponding quantized enveloping algebra., Comment: 36 pages, Section 5 and more references added

Let G be a split semisimple algebraic group over a field k and let X be the flag variety (i.e., the variety of Borel subgroups) of G twisted by a generic G-torsor. We start a systematic study of the conjecture, raised in [5] in form of a question, that the canonical epimorphism of the Chow ring of X onto the associated graded ring of the topological filtration on the Grothendieck ring of X is an isomorphism. Since the topological filtration in this case is known to coincide with the computable gamma filtration, this conjecture indicates a way to compute the Chow ring. We reduce its proof to the case of k=Q. For simply-connected or adjoint G, we reduce the proof to the case of simple G. Finally, we provide a list of types of simple groups for which the conjecture holds. Besides of some classical types considered previously (namely, A, C, and the special orthogonal groups of types B and D), the list contains the exceptional types G2, F4, and simply-connected E6.

Let $G$ be the group of $\mathbb R$--points of a semisimple algebraic group $\mathcal G$ defined over $\mathbb Q$. Assume that $G$ is connected and noncompact. We study Fourier coefficients of Poincar\' e series attached to matrix coefficients of integrable discrete series. We use these results to construct explicit automorphic cuspidal realizations, which have appropriate Fourier coefficients $\neq 0$, of integrable discrete series in families of congruence subgroups. In the case of $G=Sp_{2n}(\mathbb R)$, we relate our work to that of Li [15]. For $\mathcal G$ quasi--split over $\mathbb Q$, we relate our work to the result about Poincar\' e series due to Khare, Larsen, and Savin [16].

This paper establishes upper and lower bounds on the speed of approximation in a wide range of natural Diophantine approximation problems. The upper and lower bounds coincide in many cases, giving rise to optimal results in Diophantine approximation which were inaccessible previously. Our approach proceeds by establishing, more generally, upper and lower bounds for the rate of distribution of dense orbits of a lattice subgroup Γ in a connected Lie (or algebraic) group G, acting on suitable homogeneous spaces G/H. The upper bound is derived using a quantitative duality principle for homogeneous spaces, reducing it to a rate of convergence in the mean ergodic theorem for a family of averaging operators supported on H and acting on G/Γ. In particular, the quality of the upper bound on the rate of distribution we obtain is determined explicitly by the spectrum of H in the automorphic representation on L 2 L^{2} (Γ ∖ \setminus G). We show that the rate is best possible when the representation in question is tempered, and show that the latter condition holds in a wide range of examples.

A closed subgroup of a semisimple algebraic group is called irreducible if it lies in no proper parabolic subgroup. In this paper we classify all irreducible subgroups of exceptional algebraic groups $G$ which are connected, closed and simple of rank at least $2$. Consequences are given concerning the representations of such subgroups on various $G$-modules: for example, with one exception, the conjugacy classes of irreducible simple connected subgroups of rank at least $2$ are determined by their composition factors on the adjoint module for $G$., 46 pages; version to appear in J. Alg

Let $G$ be a semisimple algebraic group defined over an algebraically closed field of characteristic 0 and $P$ be a parabolic subgroup of $G$. Let $M$ be a $P$-module and $V$ be a $P$-stable closed subvariety of $M$. We show in this paper that if the varieties $V$ and $G\cdot M$ have rational singularities, and the induction functor $R^i\text{ind}_P^G(-)$ satisfies certain vanishing condition then the variety $G\cdot V$ has rational singularities. This generalizes the main result of Kempf in [Invent. Math., 37 (1976), no. 3]. As an application, we prove the property of having rational singularities for nilpotent commuting varieties over $3\times 3$ matrices., Title and abstract were changed; the method was generalized to semisimple algebraic groups

A key result in a 2004 paper by S. Arkhipov, R. Bezrukavnikov, and V. Ginzburg (ABG) gives an equivalence of the bounded derived category of finite dimensional modules for the principal block of a Lusztig quantum algebra at an $\ell^{th}$ root of unity, with am explicit full subcategory of the bounded derived category of integrable type 1 modules for a Borel part of the quantum algebra. Some restrictions on $\ell$ are required; in particular, it is assumed $\ell > h$, the Coxeter number. The same paper suggests there is an analogous result for representations of semisimple algebraic groups in characteristic $p>0$, and the authors of this paper have proved such a result (with $p>h$) in a separate manuscript, recently posted. The philosophy of the proof is a variation on that of ABG, but contains new ingredients and some missing details, even in the quantum case. The present paper continues the study of the modular case, showing the equivalence constructed (via a right derived functor of induction from a Borel part) behaves well, when $p>2h-2$, with respect to certain weight poset "truncations", making use of van der Kallen's 1989 "excellent order" highest weight categories. This implies, in particular, that the equivalence can be reformulated in terms of triangulated categories associated to derived categories of finite dimensional quasi-hereditary algebras. We expect that a similar result holds in the quantum case., 10 pages

If G is a complex semisimple algebraic group, we characterize the normality and the smoothness of its simple linear compactifications, namely those equivariant GxG-compactifications which possess a unique closed orbit and which arise in a projective space of the shape P(End(V)), where V is finite dimensional rational G-module. Both the characterizations are purely combinatorial and are expressed in terms of the highest weights of V. In particular, we show that Sp(2r) (with r > 0) is the unique non-adjoint simple group which admits a simple smooth compactification., v2: minor changes, final version. To appear in Math. Z

Following the program of algebraic Frobenius splitting begun by Kumar and Littelmann, we use representation-theoretic techniques to construct a Frobenius splitting of the cotangent bundle of the flag variety of a semisimple algebraic group over an algebraically closed field of positive characteristic. We also show that this splitting is the same as one of the splittings constructed by Kumar, Lauritzen, and Thomsen., Bug fixes; final version. To appear in J. Algebra

We introduce a new construction of exceptional objects in the derived category of coherent sheaves on a compact homogeneous space of a semisimple algebraic group and show that it produces exceptional collections of the length equal to the rank of the Grothendieck group on homogeneous spaces of all classical groups., v1: 51 page; v2: 55 pages, to appear in JEMS

An element w of a Weyl group W is called elliptic if it has no eigenvalue 1 in the standard reflection representation. We determine the order of any representative g in a semisimple algebraic group G of an elliptic element w in the corresponding Weyl group W. In particular if w has order d and G is simple of type different from C_n or F_4, then g has order d in G., 20 pages

For all spherical homogeneous spaces G/H, where G is a simply connected semisimple algebraic group and H a connected solvable subgroup of G, we compute the spectra of the representations of G on spaces of regular sections of homogeneous line bundles over G/H., v2: 14 pages, minor corrections

Let $\sigma_1$ and $\sigma_2$ be commuting involutions of a semisimple algebraic group $G$. This yields a $Z_2\times Z_2$-grading of $\g=\Lie(G)$, $\g=\bigoplus_{i,j=0,1}\g_{ij}$, and we study invariant-theoretic aspects of this decomposition. Let $\g$ be the $Z_2$-contraction of $\g$ determined by $\sigma_1$. Then both $\sigma_2$ and $\sigma_3:=\sigma_1\sigma_2$ remain involutions of the non-reductive Lie algebra $\g$. The isotropy representations related to $(\g, \sigma_2)$ and $(\g, \sigma_3)$ are degenerations of the isotropy representations related to $(\g, {\sigma_2})$ and $(\g, {\sigma_3})$, respectively. We show that these degenerated isotropy representations retain many good properties. For instance, they always have a generic stabiliser and their algebras of invariants are often polynomial. We also develop some theory on Cartan subspaces for various $Z_2$-gradings associated with the $Z_2\times Z_2$-grading of $\g$., Comment: 31 pages

It is well-known that the Eulerian polynomials, which count permutations in S_n by their number of descents, give the h-polynomial/h-vector of the simple polytopes known as permutohedra, the convex hull of the Sn -orbit for a generic weight in the weight lattice of Sn . Therefore the Eulerian polynomials give the Betti numbers for certain smooth toric varieties associated with the permutohedra. In this thesis we derive recurrences for the h-vectors of a family of polytopes generalizing this. The simple polytopes we consider arise as the orbit of a non-generic weight, namely a weight fixed by only the simple reflections J = {sn , sn−1 , · · · , sn−k+1 } for some k with respect to the An root lattice. Furthermore, they give rise to certain rationally smooth toric varieties X(J) which comes naturally from the theory of algebraic monoids. Using effectively the cross section lattice of a reductive monoid and the combinatorics of simple polytopes, we obtain a recurrence formula for the Poincar e polynomial of X(J) in terms of the Eulerian polynomials. Furthermore, we compute explicitely the Poincar ́ polynomial of X(J) using the method of descent systems in the case of (W, S) finite Weyl group of type An and J combinatorially smooth of the following forms: 1. J = {s_1 , s_4 , s_5 , · · · , s_n } ⊂ S 2. J = {s_4 , s_5 , · · · , s_n } ⊂ S.

Let G be a simply connected semisimple algebraic group over an algebraically closed field k of positive characteristic. We will untwist the structure of G-modules by a newly found splitting of the Frobenius endomorphism on the algebra of distributions of G, which, for example, explains the subsumed G-equivariant nature of the Frobenius splitting of the flag variety of G., Comment: 23 pages

The extended weight semigroup of a homogeneous space G/H of a connected semisimple algebraic group G characterizes the spectra of the representations of G on the spaces of regular sections of homogeneous linear bundles over G/H, including the space of regular functions on G/H. We compute the extended weight semigroups for all strictly irreducible affine spherical homogeneous spaces G/H, where G is a simply connected non-simple semisimple complex algebraic group and H a connected closed subgroup of it. In all the cases we also find the highest weight functions corresponding to the indecomposable elements of this semigroup. Among other things, our results complete the computation of the weight semigroups for all strictly irreducible simply connected affine spherical homogeneous spaces of semisimple complex algebraic groups., v2: 23 pages, cosmetic changes

We give a cohomological criterion for existence of outer automorphisms of a semisimple algebraic group over an arbitrary field. This criterion is then applied to the special case of groups of type D_2n over a global field, which completes some of the main results from the paper "Weakly commensurable arithmetic groups and isospectral locally symmetric spaces" (Pub. Math. IHES, 2009) by Prasad and Rapinchuk and gives a new proof of a result from another paper by the same authors., v2: Small changes and new title since v1 // v3: Expanded the introduction

In this paper we prove that if G is a connected, simply-connected, semi-simple algebraic group over an algebraically closed field of sufficiently large characteristic, then all the blocks of the restricted enveloping algebra (Ug)_0 of the Lie algebra g of G can be endowed with a Koszul grading (extending results of Andersen, Jantzen and Soergel). We also give information about the Koszul dual rings. Our main tool is the localization theory in positive characteristic developed by Bezrukavnikov, Mirkovic and Rumynin., Comment: 85 pages, final (slightly condensed) version, to appear in DMJ

For a connected complex semi-simple Lie group $G$ and a fixed pair $(B, B^-)$ of opposite Borel subgroups of $G$, we determine when the intersection of a conjugacy class $C$ in $G$ and a double coset $BwB^-$ is non-empty, where $w$ is in the Weyl group $W$ of $G$. The question comes from Poisson geometry, and our answer is in terms of the Bruhat order on $W$ and an involution $\mc \in W$ associated to $C$. We study properties of the elements $\mc$. For $G = SL(n+1, \Cset)$, we describe $\mc$ explicitly for every conjugacy class $C$, and for the case when $w \in W$ is an involution, we also give an explicit answer to when $C \cap (BwB)$ is non-empty., 20 pages. Revised version. to appear in Transformation Groups

Let $\sigma$ be a simple involution of an algebraic semisimple group $G$ and let $H$ be the subgroup of $G$ of points fixed by $\sigma$. If the restricted root system is of type $A$, $C$ or $BC$ and $G$ is simply connected or if the restricted root system is of type $B$ and $G$ is adjoint, then we describe a standard monomial theory and the equations for the coordinate ring $k[G/H]$ using the standard monomial theory and the Pl\"ucker relations of an appropriate (maybe infinite dimensional) Grassmann variety., Comment: 48 pages

Let G be a simply connected semisimple algebraic group and let H0 be the subgroup of points fixed under an involution of G. If V is an irreducible representation with a line L of vectors fixed by H0 we consider the closure of the G-orbit of L in \( \mathbb{P}{\left( V \right)} \). We describe the G-orbits of this closure and we prove that the normalization of this variety is homeomorphic to the variety itself.

We prove that an irreducible lattice in a semisimple algebraic group is virtually isomorphic to an arithmetic lattice if and only if it admits a faithful self-similar action on a rooted tree of finite valency., Comment: 6 pages

Let G be an arithmetic lattice in a semisimple algebraic group over a number field. We show that if G has the congruence subgroup property, then the number of n-dimensional irreducible representations of G grows like n^a, where a is a rational number.

We prove some general results on the T-equivariant K-theory K T (G/P) of the flag variety G/P, where G is a complex semisimple algebraic group, P is a parabolic subgroup, and T is a maximal torus contained in P. In particular, we make a conjecture about a positivity phenomenon in K T (G/P) for the product of two basis elements written in terms of the basis of K T (G/P) given by the dual of the structure sheaf (of Schubert varieties) basis. (For the full flag variety G/B, this dual basis is closely related to the basis given by Kostant-Kumar.) This conjecture is parallel to (but different from) the conjecture of Griffeth-Ram for the structure constants of the product in the structure sheaf basis. We give explicit expressions for the product in the T-equivariant K-theory of projective spaces in terms of these bases. In particular, we establish our conjecture and the conjecture of Griffeth-Ram in this case.

Let $H$ be a semisimple algebraic group. We prove the semistable reduction theorem for $\mu$--semistable principal $H$--bundles over a {\it smooth projective variety $X$} defined over the field $\bc$. When $X$ is a {\it smooth projective surface} and $H$ is simple, we construct the algebro--geometric Donaldson--Uhlenbeck compactification of the moduli space of $\mu$--semistable principal $H$--bundles with fixed characteristic classes and describe its points. For large characteristic classes we show that the moduli space of $\mu$--stable principal $H$--bundles is non--empty., Comment: 46 pages

Let $G$ be a simply connected semisimple algebraic group and let $K$ be the subgroup of points fixed by an involution of $G$. For certain representations containing a line $r$ preserved by $K$, we study the normality of the closure of the set of vectors which are $G$ conjugate to a vector in $r$. Some applications of our result to the normality of certain classical varieties are given.

We define and study a family of partitions of the wonderful compactification \bar{G} of a semi-simple algebraic group G of adjoint type. The partitions are obtained from subgroups of G \times G associated to triples (A_1, A_2, a), where A_1 and A_2 are subgraphs of the Dynkin graph \Gamma of G and a : A_1 \to A_2 is an isomorphism. The partitions of \bar{G} of Springer and Lusztig correspond respectively to the triples (\emptyset, \emptyset, \id) and (\Gamma, \Gamma, \id)., Comment: 24 pages, AMS Latex, minor revisions in v2

International audience; Let GF denote the rational points of a semisimple group G over a non-archimedean local field F, with Bruhat-Tits building X. This paper contains five main results. We prove a convergence theorem for sequences of parahoric subgroups of GF in the Chabauty topology, which enables to compactify the vertices of X. We obtain a structure theorem showing that the Bruhat-Tits buildings of the Levi factors all lie in the boundary of the compactification. Then we obtain an identification theorem with the polyhedral compactification (previously defined in analogy with the case of symmetric spaces). We finally prove two parametrization theorems extending the BruhatTits dictionary between maximal compact subgroups and vertices of X: one is about Zariski connected amenable subgroups, and the other is about subgroups with distal adjoint action.

Let G be a connected simple semisimple algebraic group over a local field F of arbitrary characteristic. In a previous article by the author the Zariski dense compact subgroups of G(F) were classified. In the present paper this information is used to give another proof of a theorem of Prasad [8] (also proved by Margulis [3]) which asserts that, if G is isotropic, every non-discrete closed subgroup of finite covolume contains the image of Open image in new window where Open image in new window denotes the universal covering of G. This result played a central role in Prasad’s proof of strong approximation. The present proof relies on some basic properties of Weil restrictions over possibly inseparable field extensions, which are also proved here.

These are the notes for the lecture given by the author at the "Current Events" Special Session of the AMS meeting in Baltimore on January 17, 2003. Topics reviewed include the Langlands correspondence for GL(n) in the function field case and its proof by V.Drinfeld and L.Lafforgue, the geometric Langlands correspondence for GL(n) and its proof by D.Gaitsgory, K.Vilonen and the author, and the work of A.Beilinson and V.Drinfled on the quantization of the Hitchin system and the Langlands correspondence for an arbitrary semisimple algebraic group., 35 pages

We prove that in positive characteristic a module with good filtration for a group of type E6 restricts to a module with good filtration for a subgroup of type F4. (Recall that a filtration of a module for a semisimple algebraic group is called good if its layers are dual Weyl modules.) Our result confirms a conjecture of Brundan for one more case. The method relies on the canonical Frobenius splittings of Mathieu. Next we settle the remaining cases, in characteristic not 2, with a computer-aided variation on the old method of Donkin., 16 pages; proof of Brundan's conjecture added

Let $ G^\tau $ be a connected simply connected semisimple algebraic group, endowed with generalized Sklyanin-Drinfel'd structure of Poisson group, let $ H^\tau $ be its dual Poisson group. By means of quantum double construction and dualization via formal Hopf algebras, we construct new quantum groups $ U_{q,\varphi}^M(\frak{h}) $ --- dual of $ U_{q,\varphi}^{M'}(\frak{g}) $ --- which yield infinitesimal quantization of $ H^\tau $ and $ G^\tau \, $, we study their specializations at roots of 1 (in particular, their classical limits), thus discovering new quantum Frobenius morphisms. The whole description dualize for $ H^\tau $ what was known for $ G^\tau $, completing the quantization of the pair $ (G^\tau,H^\tau) $., Comment: AMS-TeX file, 45 pages, final author's version of the printed paper

Let G be a connected and simply connected semisimple algebraic group over an algebraically closed field k of characteristic p > 0, T a maximal torus of G and B a Borel subgroup containing T . Each weight in X(T ) determines a line bundle on the flag varietyG/B. It turns out that cohomology of the line bundle is isomorphic to the derived functor of the induction functors from the category of B-modules to the category of G-modules for each 1-dimensional B-module defined by a weight in X(T ). Further, IndGBλ = H (G/B, λ) turns out to be the dual of a Weyl module. One of the main problems is to calculate the characters of the irreducible G-modules. For p = 0, the character of the irreducible G-module of highest weight λ is given by Weyl’s character formula and the G-module structures of the cohomology of line bundles are well understood. However, for p > 0, the story is quite different and many of the results remain conjectural. Since characters of Weyl modules are given by Weyl’s character formula, understanding the structure ofH(G/B, λ) turns out to be the main problem. It is also interesting to understand the structure of the higher cohomology, which might help us to understand H. H. Andersen has a series of papers toward the understanding of the higher cohomology, such as the simple socle of H and filtrations of H i [4]. In [5] , using the representations of infinitesimal subgroup schemes of G, he proved generically that H(G/B,w ·λ) has simple socle and simple head and their highest weights can be calculated. In [6], he proved that the socle series of H(λ) comes generically from the

Let G be a connected semisimple algebraic group over a field of non-zero characteristic p, with a maximal torus T. On G we have the Frobenius endomorphism F induced by the pth power map on the underlying field. The (scheme-theoretic) kernel of F is the first Frobenius kernel G,, and the inverse image of T under F is the “Frobenius subgroup” G, T. The purpose of this paper is to study certain questions in the (rational) representation theory of G, T. Let B be a Bore1 subgroup of G which contains T. In [6], Cline, Parshall, and Scott obtained a result which describes the functor “induction from B to G” as the composite of functors of the form “induction from B to a minimal parabolic subgroup P” over a sequence of minimal parabolics coming from a given reduced expression of the long word in the Weyl group. In Section 5, we obtain the analogous result for G, T. In Section 6, we develop a theory of intertwines for a certain class of G, T-modules. This gives a direct proof of an extended strong linkage principle for G, T, independent of the strong linkage principle for G. (Originally, Jantzen [17] proved that strong linkage for G, T, and more generally for G, T, n > 1, follows from the strong linkage principle for G. 1 Recently we have applied this method to give a new proof of an extended strong linkage principle for G; see [ 121. From the results of Section 5 we can write down a formula for the image of one of the intertwines in our class of modules. (This also follows from results of Jantzen [16, 171.) Composing the intertwines in question, relative to the class of modules corresponding to a given character of T, gives a map whose image is the irreducible G, T-module of that highest weight. In case the character is a restricted dominant character, the image coincides with the irreducible G-module of that highest weight, by a result

Let G be a semisimple algebraic group scheme. defined and split over the prime subfield of an algebraically closed field k of characteristic P > 0. if F: G + G is the Frobenius morphism, and T is a maximal split torus in G, then TG,. is the inverse image of T under F’. Thus TG’, is a non-reduced subgroup scheme of G. Let B be a Bore1 subgroup of Gz containing 7. whose opposite B” determines the positive system of roots in the root system of G relative to T. Let TB, = B xc; TG, = B n TG,, the scheme theoretic intersection of B with 7-G‘,. It is well known [J2, Sect. 3.1 that injective ‘TG,-mods( as a factor of Q,(p), coincides with the multiplicity of L,(p) as a composition factor in lnd$;(i,). At the same time, if i is generic relative to an alcove in the sense defined in [CPS 1, Appendix A] the multiplicities of L,(p) in Tndyi;(/I) coincide with the multiplicities [IndE(/i): L(p)]. where L.(/f) denotes the rational irreducible G-module of high weight /i relative to B”. Thus, in the generic case, one can translate the problem of computing multiplicities of irreducibles in Indg(/l) to 2 question about the structure of rational injectivc TG,-modules. We begin, here, a study of the structure of rational injectivc 7%,-modules from the point of view of the parabolic subgroup schemes of 7’G,. In this, we proceed by analogy with the early work [H] of Harish-Chandra, [ Sp] of Springer, and with the more recent work [RS] of Ronan and Smith. If P is a parabolic subgroup scheme of G, containing T. let TP, = P xc; TG, = P n TG,. Then TP, has a Levi factorization: TP, = TL, x lip,, where C:,,, denotes the unipotcnt radical of TP,, and TL, its Levi factor

We describe a way to compute the p-relative version of the Berhuy–Reichstein canonical dimension for an arbitrary split semisimple algebraic group over an arbitrary field of an arbitrary characteristic (p is any prime integer). The canonical p-dimension is computed for all split simple groups of classical types.