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Cabanes M., Enguehard M., Bollobas B. (Ed) — Representation Theory of Finite Reductive Groups (New Mathematical Monographs Series), Vol. 1
Cabanes M., Enguehard M., Bollobas B. (Ed) — Representation Theory of Finite Reductive Groups (New Mathematical Monographs Series), Vol. 1

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Название: Representation Theory of Finite Reductive Groups (New Mathematical Monographs Series), Vol. 1

Авторы: Cabanes M., Enguehard M., Bollobas B. (Ed)


At the crossroads of representation theory, algebraic geometry and finite group theory, this book blends together many of the main concerns of modern algebra, synthesising the past 25 years of research, with full proofs of some of the most remarkable achievements in the area. Cabanes and Enguehard follow three main themes: first, applications of étale cohomology, leading to the proof of the recent Bonnafé-Rouquier theorems. The second is a straightforward and simplified account of the Dipper-James theorems relating irreducible characters and modular representations. The final theme is local representation theory. One of the main results here is the authors' version of Fong-Srinivasan theorems. Throughout the text is illustrated by many examples and background is provided by several introductory chapters on basic results and appendices on algebraic geometry and derived categories. The result is an essential introduction for graduate students and reference for all algebraists.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2004

Количество страниц: 436

Добавлена в каталог: 27.05.2008

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Предметный указатель
$(a, b)_G$, a scalar product      xvii
$(\Lambda, K, k)$, l-modular splitting system      75
$(\LambdaG)_L$      59
$(\mathcal{O}, K, k), l-modular splitting system      xvii
$*R^G_L$, Harish — Chandra restriction      47
$A_n$, $BC_n$, $D_n$, (types of Coxeter groups)      24
$A_n$, $B_n$, $C_n$, $D_n$, E, $F_4$, $G_2$, (types of crystallographic root systems)      119
$a_n$, ($n \in N)$      90
$a_w$, ($w\in W)$, standard basis element of Hecke algebra      41 44
$a_{g,\tau,\sigma,}$ morphism between induced cuspidal triples      8
$bc_n$      24
$Br_P$, Brauer morphism      76
$B_G(M)$, $b_G(M)$, block, or block idempotent, of G acting by Id on the indecomposable module M      77
$B_G(\chi)$, block of G not annihilated by $\chi \in Irr(G)$      75
$b_l(G^F, s)$, sum of l-block idempotents defined by $\mathcal{E}_l(G^F, s)$      134
$B_w(w \in W)$      30
$B_{\delta}$, $B_w$, (B in a BN-pair, $\delta\in \Delta$, $w \in W)$, subgroups of B      30
$C \mapsto C[n]$, ($n \in \mathbb{Z})$ shift on complex C      57 375
$C((M_{\sigma}, f_{\sigma}^{\sigma}))      58
$CSO^+_{2m} = CSO_{2m,0}$, $CSO^-_{2m} = CSO_{2m,w}$      240 320
$cusp_k(\mathcal{L})$, ($\mathcal{L}$ a set of subquotients), set of cuspidal triples      7
$C^0_G(g)$, the connected centralizer of g in G      393
$C^{\epsilon}(\mathcal{A}), (\epsilon\in {—, +, b, \empty})$, categories of complexes on the abelian category $\mathcal{A}$      375
$C_H(X)$, centralizer of X in H      xv
$Dec_A (M)$, decomposition matrix of the A-module M      84
$deg(\sigma)$, degree of the simplex $\sigma$      58
$D^b(A) = D^b(A - mod)$      381
$d^{x,G}$, $d^x$, generalized decomposition map for $x\in G_l$      74 77
$D^{\epsilon}(A), (\epsilon \in {-, +, b, \empty})$, derived categories of the abelian category A      377
$D^{\epsilon}(X), (\epsilon \in {-, +, b, \empty})$subcategory of $D^{\epsilon}(Sh_A(X_{et}))$      407
$D_I$, $D_{IJ}$, with $I, J \in \Delta$, sets of distinguished elements of W      24
$D_{(G)}$, duality functor      67
$e_l^{G^F}, some central idempotent of $KG^F$      134
$e_{\chi}$, central primitive idempotent of K G for $\chi\in Irr(G)$      132
$f_*$, $f^*$, direct and inverse images      383 399 406
$G^0$      393
$G_f$, $G_b$, subgroups of $G = G_aG_b$      348
$G_{ss}$      393
$G_{\pi}$, $G_l$, set of $\pi$- or l-elements      xv
$H \lhd G$, H is a normal subgroup of G      xv
$H^i (C)$, i-th cohomology group of a cochain complex C      376
$H^i (X, \mathcal{F})$, ($\mathcal{F}$ a sheaf on $X_{et}$), ith cohomology group of a sheaf      385
$H^i (X, \mathcal{F})$, ($\mathcal{F}$ a sheaf on $X_{et}$), ith cohomology group with compact support      409
$H_i(C)$, i-th homology group of a chain complex C      57
$I'_{\sigma}$      322
$Ind^B_A$, induction from a subalgebra      289
$Ind^G_{(P,V)}$, induction from a subquotient      3
$Ind_H^G$, induction from a subgroup      xvi
$inj_{\mathcal{A}}$      378
$I^G_{\tau}$      112
$j_A^B$      168
$K^{\epsilon}(A)$, homotopic category      376
$L_I$, Levi subgroup defined by $I \subset \cong \Delta$      32
$Mat_n(A)$, matrix algebra      xvi
$M_X$, constant sheaf associated with X      406
$M_{\sigma}$      322
$N_h (X)$, normalizer of X in a group H      xv
$N_w$, with $w \in W$, a map $Y(T) \rightarrow T^{wF}$      123
$n_{\pi}$, the $\pi$-part of integer n      xv
$part(\pi-)      xv
$pr_E$      132
$P_I$, parabolic subgroup defined by $I\subset S \leftrightarrow \Delta$      28
$P_{g, f}$, polynomial order      190
$reg_G$, regular character of G      132
$Res^G_{(P, V)}$, restriction to a subquotient      5
$Res_H^G$, restriction to a subgroup      xvi
$R\Gamma(X,\mathcal{F})$, $R_c\Gamma(X,\mathcal{F})$      409
$R^G_L$, Harish — Chandra induction for L a Levi subgroup of G      47
$R^G_T\theta$, Deligne — Lusztig character      126
$R_cf_*$, direct image with compact support      408
$R_l^GB_l$, twisted induction of blocks      337
$R_u$      394
$R_{L\subset P}^G$      125
$R_{l\subset P}^G$, Deligne-Lusztig twisted induction      125
$Sh_A(X_{et})$      405
$SL_n$      28
$SO^+_{2m} = SO_{2m,0}$      29 222
$SO_{n,v} (v = 0, w      1 d)$
$Spin_{2m,v}$      228
$Sp_{2m}$      38
$St_{C^F}$, the Steinberg module or character      301
$S_w$      177
$S_{(w,\theta)}$      150
$S_{\mathcal{O}}(n, q^a)$ (q-Schur algebra)      326
$Tr_Q^P$, relative trace      76
$t_i$      279
$U>\lhd T$, a semi-direct product      xv
$U_I$, a subgroup of U, defined by $I\subset S\cong\Delta$      30
$v(\delta, I)$, where $I \subset \Delta$ and $\delta \in \Phi \,\ I$      26 47 48
$v_m^{(n)}$      281
$W_I$, a subgroup of W, defined by $I\subset S\cong\Delta$      23
$w_I$, with $I\subset S\cong\Delta$      26
$w_m^{(n)}$,      281
$w_{\theta}$, with $\theta$ a linear character of $T^{wF}$      149 168
$X\stackrel{w}{\longrightarrow}Y      146
$x_I$, ($I\subset S$)      273
$X_V$, $X_V^{G,F}$, subvariety of G/P      110
$X_{et}, the etale topology on X      404
$X_{I,v}$      147
$X_{\alpha}$, $\alpha$ root      30
$x_{\lambda}$      276
$X_{\sigma}$      322
$y_I$      273
$Y_V$, $Y_V^{G,V}$, subvariety of G/V      110
$Y_{I,v}$      147
$Y_{[v,w]}$      162
$y_{\lambda}$      276
$\bar{X}(w)$      112
$\beta * \gamma$      78
$\beta$-set      78
$\beta(\lambda)$, $\lambda$ a partition      78
$\cap\downarrow$, non-symmetric intersection of subquotients      6
$\chi(\sigma_))$      59
$\chi_C$, (C a complex of A-modules), Lefschetz character      381
$\Delta$, a set of simple roots in $\Phi$, in bijection with S      23
$\delta_\pi$      135
$\Delta_{\lambda}$      91
$\epsilon(\lambda, \gamma)$      79
$\epsilon(\sigma)$      78
$\epsilon_g$, a sign      126
$\epsilon_m$      285
$\eta_{w,j}$      162
$\Gamma(U, \mathcal{F}) = \mathcal{F}(U)$      382
$\Gamma_{G^F}$, $\Gamma_{G^F,1}$, Gelfand — Graev module      301
$\lambda *$, dual partition      276
$\Lambda$-regular set of subquotients      6
$\lambda\gg \mu$      314
$\lambda\ll \mu$      276
$\Lambda_{\sigma}$      322
$\mathbb{A}_F^n$      391
$\mathbb{A}_F^{sh}$      418
$\mathbb{G}_a$, $\mathbb{G}_m$, additive and multiplicative algebraic groups      393
$\mathbb{P}_F^n$      391
$\mathcal{D}(\mathcal{F})$      413
$\mathcal{E}(G^F, s)$ (s a semi-simple rational element in G*), a rational series      127
$\mathcal{E}(kG, \tau)$      18
$\mathcal{E}_l(G^F, s)$, a union of rational series (and of blocks)      133
$\mathcal{F}[x, y](\theta)$      168
$\mathcal{F}^+$, sheafification of a presheaf      383 405
$\mathcal{F}_w(t)$      150
$\mathcal{H}_k(G, U)$      88
$\mathcal{H}_R(BC_n, Q, q)$      279
$\mathcal{H}_R(G, B) = End_{RG}(Ind_B^GR)$      44
$\mathcal{H}_R(W, (q_s))$, Hecke algebra with parameters $q_s(s\in S)$      44
$\mathcal{H}_{$\mathcal{O}}(\mathfrak{C}n, q)$      275
$\mathcal{H}_{\sigma}$      322
$\mathcal{L}$, Levi system of subquotients      32
$\mathcal{L}_{X/G} (M)$, (M a G-module, X/G a quotient variety), a coherent $\mathcal{O}_{x/g}$ -module      400
$\mathcal{O}_X$, structure sheaf of a scheme      390
$\mathcal{O}_X$-module      399
$\mathcal{R}_w$      177
$\mathcal{T}f_x$, tangent map at x      392
$\mathcal{T}X$, (X a variety), tangent sheaf      392
$\mathfrak{S}_n$      23
$\mathfrak{S}_X$      78
$\Omega(V)$      228
$\Omega_{2n}$      219
$\otimes^L_A$, left derived tensor product      381
$\Phi$, $\Phi^+$, $\Phi^-$      23
$\phi_E$ -subgroup      190
$\phi_n$, cyclotomic polynomial      xvi
$\Phi_w$,      24
$\pi$-regular element      xv
$\pi(n)$, $\pi_d(n)$, numbers of partitions      309
$\pi_1(X, \bar{x})$, $\pi_1(X)$, fundamental group      416
$\pi_1^D(X)$, tame fundamental group with respect to D      417
$\pi_1^t(X, \bar{x})$, tame fundamental group      416
$\pi_m$, $\tilde{\pi}_m$      281
$\psi(\lambda, I)$      91
$\Theta(G, F)$, $\Theta_k(G, F)$, $\Theta(G, F, s)$      149
$\tilde{\mathcal{E}}(G^F, s)$ (s a semi-simple rational element in G*), a geometric series      127
(q-)Schur algebra      298
(W, S), a Coxeter system      23
4\RHO_{c, d}(\lambda)$      309
A — Mod, A — mod, module categories      xvi
A(t)      214
Abelian defect conjecture (Broue)      369
Abelian, additive categories      374
Acyclic      376
Alperin’s “weight” conjecture      96
Ariki-Koike      296
B(s)      214
Base change      398
Base change for a proper morphism      408
Basic set of characters      201
Bi-functor      380
Bi-partition      330
Bi-projective      57
Bimodule      xvi
Block (l-), l-block idempotent (l a prime number)      75
BN-pair      27
BN-pair, split, of characteristic p      30
BN-pair, strongly split,      30
Bonnafe — Rouquier’s theorem      141
Borel subgroup      394
Brauer morphism      76
Brauer’s “second Main Theorem” and “third Main Theorem”      77
Broue — Michel’s theorem      131
Broue’s abelian defect conjecture      369
Building      40
CF(G, A), space of central functions from G to A      xvii
CF(G, K, B), CF(G, B), space of central functions defined by the block B of G      75
CL(V), Clifford group      228
Coefficient system      58
Coherent sheaf      399
Compact support, cohomology with, direct image with      408
Compactification      408
Complex, acyclic      57 376
Complex, bounded      57
Complex, perfect      381
Complex, tensor product of,s      57
Composition of an integer, $\lambda\models n$      308
Cone(f)      375
Conformal groups      240
Control subgroup      363
Core      79
CSP      320
Cuspidal module, triple      7
Dade’s conjectures      100
DC, duality complex      64
Dec(A), decomposition matrix of the algebra A      84
Decomposition matrix      84
Defect, central      77
Defect, zero      77
Degenerate symbols      216
Deligne — Lusztig      xii
Derived categories      377
Derived functors      379
Dimension of a point      398
Dimension of a variety      391
Dipper — Du      298
Dipper — James      271
Divisor, smooth      392
Divisor, smooth, with normal crossings      392
Donovan’s conjecture      370
Duality, Alvis — Curtis      67
Duality, local      420
Duality, Poincare — Verdier      412
e(V), idempotent defined by the subgroup V      5
e-cuspidal, E-cuspidal      335
Equivalence relation between subquotients      6
Equivalence, derived      57
Equivalence, Morita      137
Etale, covering      405
Etale, morphism      404
Etale, neighborhood      405
Etale, topology      404
F, an algebraic closure of the field $\mathbb{F}_q$      103
F, an endomorphism of G      104 121 395
Fibered product      398
Fong-Srinivasan      xii
Frobenius, algebra      12
Frobenius, morphism      395
F[ V], regular functions on an F-variety V      389
G*, a group in duality with G      123
G, $G_{ad}$, $G_{sc}$, a connected reductive group over F, the associated adjoint and simply connected groups      394
Galois covering      416
Geck — Hiss — Malle      298
Generic block      358
GL      xvi
Global sections      382
Global sections, generated by      400
Grothendieck      xii 55 101 102 373 388 396 403 404 416
Group, $\pi-$      xv
Group, algebraic      393
Group, defect      76
Group, F-      393
Group, finite reductive      xiii
Group, fundamental, tame fundamental      416
Group, Grothendieck      173
Group, reductive      394
Gruber — Hiss      318
GUn      29
H or $H_Y = Hom_A (Y, -)$      15 298
Haastert      117
Harish-Chandra      40
hd(M), head of a module      xvi
Hecke algebra      2 4 44
Henselian      411
Hoefsmit’s matrices      279
Hom      385 399
Homgr      376
Homotopic category      376
Hook      78
I(v, w)      162
IBr(G), set of irreducible Brauer characters      299
Immersion      398
ind(w), ($w\in W)$, index      42
Induction, Harish-Chandra      1
Induction, twisted (or Deligne — Lusztig)      125
Intersect transversally      392
Irr(G), set of irreducible characters      xvii
Irr(G, b), subset of Irr(G) defined by a block idempotent b of G      75 132
J(A), Jacobson radical of A      xvi
Jordan decomposition of elements in G      393
Jordan decomposition of irreducible representations of $G^F$      209
Kunneth formula      412
l(w), length of$w\in W$ with respect to S      397
l-adic sheaf, l-adic cohomology      409
l-modular splitting system      xvii
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