Index of notation

The reference numbers indicate respectively the chapter, paragraph and number.

$k:$ VII.Conventions.

$V_{\lambda}$(S), $V_{\lambda}(s), V^{\lambda}$(S), $V^{\lambda}(s):$ VII.1.1.

$\mathfrak{g}_{\lambda}(\mathfrak{h}),\mathfrak{g}^{\lambda}(\mathfrak{h}):$ VII.1.3.

$a_i(x)$, rk($\mathfrak{g}$)$:$ VII.2.2.

Aut($\mathfrak{g}$), Aut$_e(\mathfrak{g}):$ VII.3.1.

$\mathscr{C}^{\infty}\mathfrak{g}=\cap \mathscr{C}^n\mathfrak{g}:$ VII.3.4.

$r_{\rho},r^0_{\rho}:$ VII.4.1.

$e(\mathfrak{g}):$ VII.5.2.

$A_V:$ VII.App.I.1.

$Df:$ VII.App.I.2.

$H, X_+, X_-:$ VIII.1.1.

$V(m):$ VIII.1.3.

$h(t),\theta (t):$ VIII.1.5.

$\mathfrak{g},\mathfrak{h},R(\mathfrak{g},\mathfrak{h}) = R:$ VIII.2.2.

$\mathfrak{g}^{\alpha},\mathfrak{h}_{\alpha},\mathfrak{s}_{\alpha}, H_{\alpha}, h_{\alpha}, \theta_{\alpha}(t), s_{\alpha}:$ VIII.2.2.

$\mathfrak{h}_{\mathbf{Q}},\mathfrak{h}^*_{\mathbf{Q}},\mathfrak{h}_{\mathbf{R}},\mathfrak{h}^*_{\mathbf{R}}:$ VIII.2.2.

$N_{\alpha \beta}:$ VIII.2.4.

$\mathfrak{g}^P,\mathfrak{h}_P:$ VIII.3.1.

$n(\alpha , \beta ), X_{\alpha}^0, H_{\alpha}^0, X_{-\alpha}^0:$ VIII.4.2.

$\theta ,\mathfrak{a},\mathfrak{a}_+,\mathfrak{a}_-:$ VIII.4.2.

$x_{\alpha \beta}, y_{\alpha \beta},\mathfrak{n}:$ VIII.4.3.

Aut($\mathfrak{g},\mathfrak{h}$)$,A(R):$ VIII.5.1.

$T_P,T_Q$, Aut$_0(\mathfrak{g})$, Aut$_0(\mathfrak{g},\mathfrak{h})$, Aut$_e(\mathfrak{g},\mathfrak{h}):$ VIII.5.2.

$(\mathfrak{g},\mathfrak{h}),R,W,B,R_+,R_-,\mathfrak{n}_+,\mathfrak{n}_-,\mathfrak{b}_+,\mathfrak{b}_-, X_{\alpha}, Y_{\alpha}, H_{\alpha}:$ VIII.6.Conventions.

$w_0:$ VIII.7.2.

$[E],[E]^*,\mathscr{R}(\mathfrak{a}):$ VIII.7.6.

$\mathbf{Z}[\Delta ], e^{\lambda}$, ch(E): VIII.7.7.

exp($\lambda$ )$:$ VIII.8.1.

$\theta (a), \theta^*(a),I(\mathfrak{g}),I(\mathfrak{g}^*),\mathbf{S}(\mathfrak{g}),\mathbf{S}(\mathfrak{g}^*):$ VIII.8.3.

$u^{\sharp}, \chi_{\lambda}, \eta , \delta , \theta , \varphi :$ VIII.8.5.

$\mathfrak{B}(\nu ),K, d:$ VIII.9.1.

$J(e^\mu):$ VIII.9.2.

$(x, h, y):$ VIII.11.1.

$x^{(n,d)}, x^{(n)},(_n^x):$ VIII.12.2.

$\mathbf{x}^{(n)},[n], c_r,\mathscr{G}:$ VIII.12.6.

$\mathscr{H},\mathscr{U}_+,\mathscr{U}_-,\mathscr{U}_0,\mathscr{L},\mathscr{G}:$ VIII.12.6.

$\mathfrak{o}(\Psi ):$ VIII.13.2.

$\mathfrak{s}\mathfrak{p}(\Psi ):$ VIII.13.3.

$G_0,C(G),D(G):$ IX.1.Conventions.

$N_G(H),Z_G(H),N(H),Z(H):$ IX.1.Conventions. $\Gamma (G):$ IX.1.2.

$\widetilde{D}(G):$ IX.2.3.

rk G: IX.2.4.

$W_G(T),W(T):$ IX.2.5.

$\mathfrak{a}_{[\mathbf{R}]},\mathfrak{g}_{(\mathbf{C})},\mathfrak{g}_{\mathbf{C}}:$ IX.3.1.

$\mathfrak{a}_u, u_{\alpha}, v_{\alpha}:$ IX.3.2.

$\mathbf{U}(\Phi ),\mathbf{S}\mathbf{U}(\Phi ),\mathfrak{u}(\Phi ),\mathfrak{s}\mathfrak{u}(\Phi ):$ IX.3.4.

$\mathbf{U}(n,\mathbf{C}),\mathbf{S}\mathbf{U}(n,\mathbf{C}),\mathfrak{u}(n,\mathbf{C}),\mathfrak{s}\mathfrak{u}(n,\mathbf{C}):$ IX.3.4. $\mathbf{O}(Q),\mathbf{S}\mathbf{O}(Q),\mathfrak{o}(Q):$ IX.3.5.

$\mathbf{O}(n,\mathbf{R}),\mathbf{S}\mathbf{O}(n,\mathbf{R}),\mathfrak{o}(n,\mathbf{R}):$ IX.3.5.

$U, V, K, \theta :$ IX.3.6.

$X(H),X(f):$ IX.4.1.

$\delta (a):$ IX.4.1.

$\Gamma (f):$ IX.4.2.

$\langle a, X\rangle :$ IX.4.2.

$\widetilde{\rho ,}\widetilde{V},\widetilde{V}_{\lambda}(G):$ IX.4.3.

$P(\rho ,T):$ IX.4.3.

$\widetilde{L}(\rho ):$ IX.4.3.

$R(G,T):$ IX.4.4.

$V(\alpha ),K(\alpha ), \iota_{\alpha}, \rho_{\alpha}:$ IX.4.4.

$Z_{\alpha},S_{\alpha}:$ IX.4.5.

$K_{\alpha}:$ IX.4.5.

$R^{\vee}(G,T):$ IX.4.5.

$N(G,T):$ IX.4.6.

$D^*(G,T),D_*(G,T):$ IX.4.9.

$D^*(f),D_*(f):$ IX.4.9.

$D^*(G),D_*(G):$ IX.4.9.

Aut(G), Aut(G$,T$)$:$ IX.4.10.

$G_r,T_r:$ IX.5.1.

$W_a,W'_a:$ IX.5.2.

$H_{\alpha ,n},\mathfrak{t}_r:$ IX.5.2.

$H_A:$ IX.5.2.

$\mathfrak{g}_{reg}:$ IX.5.2.

$f, f_r:$ IX.5.4.

$\varphi , \varphi_r, \varphi_A:$ IX.5.4.

$\mathfrak{g}_r, \psi_r:$ IX.5.4.

$w(G):$ IX.6.Conventions.

$u\cap v:$ IX.6.1.

$\omega_{G/T}\cap \omega_T:$ IX.6.2.

Ad$_{\mathfrak{g}/\mathfrak{t}}(t), \delta_G(t):$ IX.6.2.

$R_+(G,T):$ IX.6.2.

$\lambda_{\mathfrak{h}}:$ IX.6.3.

$\pi_{\mathfrak{g}}:$ IX.6.3.

$\mathscr{S}^r(X;E):$ IX.6.4.

$s^{\sharp}:$ IX.6.4.

$\Omega (X), \Omega (X)^G:$ IX.6.5.

$\Gamma (T)_{++},X_+,R_+,R_-:$ IX.7.Conventions. $X_{++}:$ IX.7.1.

$\varpi_{\alpha}, \rho :$ IX.7.1.

$\Theta (G):$ IX.7.2.

R(G): IX.7.3.

$[\tau ]$, Ch($\tau$ )$:$ IX.7.3.

$Z\Theta (G):$ IX.7.3.

$^wf,J(f):$ IX.7.4.

$\chi_{\lambda}:$ IX.7.4.

ZL$^2(G):$ IX.7.4.

$Q_{\Gamma},\widetilde{\Gamma}(\tau ):$ IX.7.6.

$\widehat{G}:$ IX.8.1.

$E_u, d(u),\|A\|_{\infty},\|A\|_2,\langle A|B\rangle :$ IX.8.1.

$F( \widehat{G}),L^2(G):$ IX.8.1.

$u(f),\overline{\mathscr{F}}(f):$ IX.8.1.

$\mathscr{F}_uA,\mathscr{F}(A):$ IX.8.1.

$\gamma (s)f, \delta (s)f:$ IX.8.1.

$L_tf,R_tf:$ IX.8.2.

$\lambda (u),\widetilde{\Gamma}(u):$ IX.8.2.

$\varphi \preccurlyeq \psi :$ IX.8.2.

$\mathscr{S}( \widehat{G}):$ IX.8.2.

$\chi_u:$ IX.8.3.

$Z\mathscr{C}^{\infty}(G),Z\mathscr{C}(G):$ IX.8.4.

$E_{(t)}:$ IX.9.4.

Spin$(n,\mathbf{R}):$ IX.3, Exerc. 5.