lie-algebras-and-their-representations
Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
coherent-families.tex (37169B)
1 \chapter{Classification of Coherent Families} 2 3 % TODO: Write an introduction 4 5 % TODOOO: Is this decomposition unique?? 6 \begin{proposition} 7 Suppose \(\mathfrak{g} = \mathfrak{s}_1 \oplus \cdots \oplus \mathfrak{s}_r\) 8 and let \(\mathcal{M}\) be a semisimple irreducible coherent 9 \(\mathfrak{g}\)-family. Then there are semisimple irreducible coherent 10 \(\mathfrak{s}_i\)-families \(\mathcal{M}_i\) such that 11 \[ 12 \mathcal{M} \cong \mathcal{M}_1 \otimes \cdots \otimes \mathcal{M}_r 13 \] 14 \end{proposition} 15 16 \begin{proof} 17 Suppose \(\mathfrak{h}_i \subset \mathfrak{s}_i\) are Cartan subalgebras, 18 \(\mathfrak{h} = \mathfrak{h}_1 \oplus \cdots \oplus \mathfrak{h}_r\) and \(d 19 = \deg \mathcal{M}\). Let \(M \subset \mathcal{M}\) be any 20 infinite-dimensional simple submodule, so that \(\mathcal{M}\) is a 21 semisimple coherent extension of \(M\). By 22 Example~\ref{thm:simple-weight-mod-is-tensor-prod}, there exists (unique) 23 simple weight \(\mathfrak{s}_i\)-modules \(M_i\) such that \(M \cong M_1 24 \otimes \cdots \otimes M_r\). Take \(\mathcal{M}_i = \mExt(M_i)\). We will 25 show that \(\mathcal{M}_1 \otimes \cdots \otimes \mathcal{M}_r\) is a 26 coherent extension of \(M\). 27 28 It is clear that \(\mathcal{M}_1 \otimes \cdots \otimes \mathcal{M}_r\) is a 29 degree \(d\) bounded \(\mathfrak{g}\)-module containing \(M\) as a submodule. 30 It thus suffices to show that \(\mathcal{M}\) is a coherent family. By 31 Example~\ref{ex:supp-ess-of-tensor-is-product}, 32 \(\operatorname{supp}_{\operatorname{ess}} (\mathcal{M}_1 \otimes \cdots 33 \otimes \mathcal{M}_r) = \mathfrak{h}^*\). To see that the map 34 \begin{align*} 35 \mathfrak{h}^* & \to K \\ 36 \lambda & \mapsto 37 \operatorname{Tr} 38 ( 39 u\! \restriction_{(\mathcal{M}_1 \otimes \cdots \otimes \mathcal{M}_r)_\lambda} 40 ) 41 \end{align*} 42 is polynomial, notice that the natural isomorphism of algebras 43 \begin{align*} 44 f : \mathcal{U}(\mathfrak{s}_1) \otimes 45 \cdots \otimes \mathcal{U}(\mathfrak{s}_1) 46 & \isoto \mathcal{U}(\mathfrak{g}) \\ 47 u_1 \otimes \cdots \otimes u_r & \mapsto u_1 \cdots u_r 48 \end{align*} 49 described in Example~\ref{ex:univ-enveloping-of-sum-is-tensor} is a 50 \(\mathfrak{g}\)-homomorphism between the tensor product of the adjoint 51 \(\mathfrak{s}_i\)-modules \(\mathcal{U}(\mathfrak{s}_i)\) and the adjoint 52 \(\mathfrak{g}\)-module \(\mathcal{U}(\mathfrak{g})\). 53 54 Indeed, given \(X = X_1 + \cdots + X_r \in \mathfrak{g}\) with \(X_i \in 55 \mathfrak{s}_i\) and \(u_i \in \mathcal{U}(\mathfrak{s}_i)\), 56 \[ 57 \begin{split} 58 f(X \cdot (u_1 \otimes \cdots \otimes u_r)) 59 & = f([X_1, u_1] \otimes u_2 \otimes \cdots \otimes u_r) 60 + \cdots 61 + f(u_1 \otimes \cdots \otimes u_{r-1} \otimes [X_r, u_r]) \\ 62 & = [X_1, u_1] u_2 \cdots u_r + \cdots + u_1 \cdots u_{r-1} [X_r, u_r] \\ 63 \text{(\([X_i, u_j] = 0\) for \(i \ne j\))} 64 & = [X_1, u_1u_2 \cdots u_r] + \cdots + [X_r, u_1 \cdots u_{r-1}u_r] \\ 65 & = [X, f(u_1 \otimes \cdots \otimes u_r)] 66 \end{split} 67 \] 68 69 Hence by Example~\ref{ex:adjoint-action-in-universal-enveloping-is-weight} 70 \(f\) restricts to an isomorphism of algebras \(\mathcal{U}(\mathfrak{s}_1)_0 71 \otimes \cdots \otimes \mathcal{U}(\mathfrak{s}_r)_0 \isoto 72 \mathcal{U}(\mathfrak{g})_0\) with image \(\mathcal{U}(\mathfrak{g})_0 = 73 \mathcal{U}(\mathfrak{s}_1)_0 \cdots \mathcal{U}(\mathfrak{s}_r)_0\). More 74 importantly, if we write \(\lambda = \lambda_1 + \cdots + \lambda_r\) for 75 \(\lambda_i \in \mathfrak{h}_i^*\) it is clear from 76 Example~\ref{ex:tensor-prod-of-weight-is-weight} that the 77 \(\mathcal{U}(\mathfrak{g})_0\)-module \((\mathcal{M}_1 \otimes \cdots 78 \otimes \mathcal{M}_r)_\lambda\) corresponds to exactly the 79 \(\mathcal{U}(\mathfrak{s}_1)_0 \otimes \cdots \otimes 80 \mathcal{U}(\mathfrak{s}_r)_0\)-module \((\mathcal{M}_1)_{\lambda_1} \otimes 81 \cdots \otimes (\mathcal{M}_r)_{\lambda_r}\), so we can see that the value 82 \[ 83 \operatorname{Tr} 84 ( 85 u_1 \cdots u_r \!\restriction_{(\mathcal{M}_1 \otimes \cdots \otimes \mathcal{M}_r)_\lambda} 86 ) 87 = \operatorname{Tr}(u_1\!\restriction_{(\mathcal{M}_1)_{\lambda_1}}) 88 \cdots 89 \operatorname{Tr}(u_r\!\restriction_{(\mathcal{M}_r)_{\lambda_r}}) 90 \] 91 varies polynomially with \(\lambda \in \mathfrak{h}^*\) for all \(u_i \in 92 \mathcal{U}(\mathfrak{s}_i)_0\). 93 94 Finally, \(\mathcal{M}_1 \otimes \cdots \otimes \mathcal{M}_r\) is a coherent 95 extension of \(M\). Since the \(\mathcal{M}_i = \mExt(M_i)\) are semisimple, 96 so is \(\mathcal{M}_1 \otimes \cdots \otimes \mathcal{M}_r\). It thus 97 follows from the uniqueness of semisimple coherent extensions that 98 \(\mathcal{M} \cong \mathcal{M}_1 \otimes \cdots \otimes \mathcal{M}_r\). 99 \end{proof} 100 101 This last result allows us to concentrate on focus exclusive on classifying 102 coherent \(\mathfrak{s}\)-families for the simple Lie algebras 103 \(\mathfrak{s}\). In addition, it turns out that very few simple algebras admit 104 irreducible coherent families at all. Namely\dots 105 106 \begin{proposition}[Fernando]\label{thm:only-sl-n-sp-have-cuspidal} 107 Let \(\mathfrak{s}\) be a finite-dimensional simple Lie algebra and suppose 108 there exists a infinite-dimensional cuspidal \(\mathfrak{s}\)-module. Then 109 \(\mathfrak{s} \cong \mathfrak{sl}_n(K)\) or \(\mathfrak{s} \cong 110 \mathfrak{sp}_{2 n}(K)\) for some \(n\). 111 \end{proposition} 112 113 \begin{corollary} 114 Let \(\mathfrak{s}\) be a finite-dimensional simple Lie algebra and suppose 115 there exists an irreducible coherent \(\mathfrak{s}\)-family. Then 116 \(\mathfrak{s} \cong \mathfrak{sl}_n(K)\) or \(\mathfrak{s} \cong 117 \mathfrak{sp}_{2n}(K)\) for some \(n\). 118 \end{corollary} 119 120 The problem of classifying the semisimple irreducible coherent 121 \(\mathfrak{g}\)-families for some arbitrary semisimple \(\mathfrak{g}\) can 122 thus be reduced to a proof by exaustion: it suffices to classify coherent 123 \(\mathfrak{sl}_n(K)\)-families and coherent 124 \(\mathfrak{sp}_{2n}(K)\)-families. We will follow this path by analysing each 125 case -- \(\mathfrak{sl}_n(K)\) and \(\mathfrak{sp}_{2n}(K)\) -- separately, 126 classifying coherent families in terms of combinatorial invariants -- as does 127 Mathieu in \cite[sec.~8,sec.~9]{mathieu}. Alternatively, Mathieu also provides 128 a more explicit ``geometric'' construction of the coherent families for both 129 \(\mathfrak{sl}_n(K)\) and \(\mathfrak{sp}_{2n}\) in sections 11 and 12 of his 130 paper. 131 132 Before we proceed to the individual case analysis, however, we would like 133 discuss some further reductions to our general problem, the first of which is a 134 crutial refinement to Proposition~\ref{thm:coherent-families-are-all-ext} due 135 to Mathieu. 136 137 % TODO: Note that we may take L(λ) with respect to any given basis 138 % TODO: Note beforehand that the construction of Verma modules and the notions 139 % of highest-weight modules in gerenal is relative on a choice of basis 140 \begin{proposition}\label{coh-family-is-ext-l-lambda} 141 Let \(\mathcal{M}\) be a semisimple irreducible coherent 142 \(\mathfrak{g}\)-family. Then there exists some \(\lambda \in 143 \mathfrak{h}^*\) such that \(L(\lambda)\) is bounded and \(\mathcal{M} \cong 144 \mExt(L(\lambda))\). 145 \end{proposition} 146 147 \begin{note} 148 I once had the opportunity to ask Olivier Mathieu himself how he first came 149 across the notation of coherent families and what was his intuition behind 150 it. Unfortunately, his responce was that he ``did not remember.'' However, 151 Mathieu was able to tell me that ``the \emph{trick} is that I managed to show 152 that they all come from simple highest-weight modules, which were already 153 well understood.'' I personally find it likely that Mathieu first considered 154 the idea of twisting \(L(\lambda)\) -- for \(\lambda\) with \(L(\lambda)\) 155 bounded -- by a suitable automorphism \(\theta_\mu : \Sigma^{-1} 156 \mathcal{U}(\mathfrak{g}) \isoto \Sigma^{-1} \mathcal{U}(\mathfrak{g})\), as 157 in the proof of Proposition~\ref{thm:coh-ext-exists}, and only after decided 158 to agregate this data in a coherent family by summing over the \(Q\)-cosets 159 \(\mu + Q\), \(\mu \in \mathfrak{h}^*\). 160 \end{note} 161 162 In case the significance of Proposition~\ref{coh-family-is-ext-l-lambda} is 163 unclear, the point is that it allows is to reduce the problem of classifying 164 the coherent \(\mathfrak{g}\)-families to that of aswering the following two 165 questions: 166 167 \begin{enumerate} 168 \item When is \(L(\lambda)\) bounded? 169 170 \item Given \(\lambda, \mu \in \mathfrak{h}^*\) with \(L(\lambda)\) and 171 \(L(\mu)\) bounded, when is \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\)? 172 \end{enumerate} 173 174 These are the questions which we will attempt to answer for \(\mathfrak{g} = 175 \mathfrak{sl}_n(K)\) and \(\mathfrak{g} = \mathfrak{sp}_{2n}(K)\). We begin by 176 providing a partial answer to the second answer by introducing an invariant of 177 coherent families, known as its \emph{central character}. 178 179 To describe this invariant, we consider the Verma module \(M(\lambda) = 180 \mathcal{U}(\mathfrak{g}) \cdot m^+\). Given \(\mu \in \mathfrak{h}^*\) and \(m 181 \in M(\lambda)_\mu\), it is clear that \(u \cdot m \in M(\lambda)_\mu\) for all 182 central \(u \in \mathcal{U}(\mathfrak{g})\). In particular, \(u \cdot m^+ \in 183 M(\lambda)_\lambda = K m^+\) is a scalar multiple of \(m^+\) for all \(u \in 184 Z(\mathcal{U}(\mathfrak{g}))\), say \(\chi_\lambda(u) m^+\) for some 185 \(\chi_\lambda(u) \in K\). More generally, if we take any \(m = v \cdot m^+ \in 186 M(\lambda)\) we can see that 187 \[ 188 u \cdot m 189 = v \cdot (u \cdot m^+) 190 = \chi_\lambda(u) \, v \cdot m^+ 191 = \chi_\lambda(u) m 192 \] 193 194 Since every highest-weight module is a quotient of a Verma module, it follows 195 that \(u \in Z(\mathcal{U}(\mathfrak{g}))\) acts on a highest-weight module 196 \(M\) of highest-weight \(\lambda\) via multiplication by \(\chi_\lambda(u)\). 197 In addition, it is clear that the function \(\chi_\lambda : 198 Z(\mathcal{U}(\mathfrak{g})) \to K\) must be an algebra homomorphism. This 199 leads us to the following definition. 200 201 \begin{definition} 202 Given a highest weight \(\mathfrak{g}\)-module \(M\) of highest weight 203 \(\lambda\), the unique algebra homomorphism \(\chi_\lambda : 204 Z(\mathcal{U}(\mathfrak{g})) \to K\) such that \(u \cdot m = \chi_\lambda(u) 205 m\) for all \(m \in M\) and \(u \in Z(\mathcal{U}(\mathfrak{g}))\) is called 206 \emph{the central character of \(M\)} or \emph{the central character 207 associated with the weight \(\lambda\)}. 208 \end{definition} 209 210 Since a simple highest-weight \(\mathfrak{g}\)-module is uniquelly determined 211 by is highest-weight, it is clear that central characters are invariants of 212 simple highest-weight modules. We should point out that these are far from 213 perfect invariants, however. Namelly\dots 214 215 % TODO: Cite the definition of the dot action 216 \begin{theorem}[Harish-Chandra] 217 Given \(\lambda, \mu \in \mathfrak{h}^*\), \(\chi_\lambda = \chi_\mu\) if, 218 and only if \(\mu \in W \bullet \lambda\). 219 \end{theorem} 220 221 This and much more can be found in \cite[ch.~1]{humphreys-cat-o}. What is 222 interesting about all this to us is that, as it turns out, central character 223 are also invariants of coherent families. More specifically\dots 224 225 \begin{proposition}\label{thm:coherent-family-has-uniq-central-char} 226 Suppose \(\lambda, \mu \in \mathfrak{h}^*\) are such that \(L(\lambda)\) and 227 \(L(\mu)\) are both bounded and \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\). 228 Then \(\chi_\lambda = \chi_\mu\). In particular, \(\mu \in W \bullet 229 \lambda\). 230 \end{proposition} 231 232 \begin{proof} 233 Fix \(u \in \mathcal{U}(\mathfrak{g})_0\). It is clear that 234 \(\operatorname{Tr}(u\!\restriction_{\mExt(L(\lambda))_\nu}) = 235 \operatorname{Tr}(u\!\restriction_{L(\lambda)_\nu}) = d \chi_\lambda(u)\) for 236 all \(\nu \in \operatorname{supp}_{\operatorname{ess}} L(\lambda)\). Since 237 \(\operatorname{supp}_{\operatorname{ess}} L(\lambda)\) is Zariski-dense and 238 the map 239 \(\nu \mapsto \operatorname{Tr}(u\!\restriction_{\mExt(L(\lambda))_\nu})\) 240 is polynomial, it follows that 241 \(\operatorname{Tr}(u\!\restriction_{\mExt(L(\lambda))_\nu}) = d 242 \chi_\lambda(u)\) for all \(\nu \in \mathfrak{h}^*\). But by the same token 243 \[ 244 d \chi_\lambda(u) 245 = \operatorname{Tr}(u\!\restriction_{\mExt(L(\lambda))_\nu}) 246 = \operatorname{Tr}(u\!\restriction_{\mExt(L(\mu))_\nu}) 247 = d \chi_\mu(u) 248 \] 249 for any \(\nu \in \operatorname{supp}_{\operatorname{ess}} L(\mu)\) and thus 250 \(\chi_\lambda(u) = \chi_\mu(u)\). 251 \end{proof} 252 253 Central characters may thus be used to distinguished between two semisimple 254 irreducible coherent families. Unfortunately for us, as in the case of simple 255 highest-weight modules, central characters are not perfect invariants of 256 coherent families: there are non-isomorphic semisimple irreducible coherent 257 families which share a common central character. Nevertheless, Mathieu was able 258 to at least establish a somewhat \emph{precarious} version of the converse of 259 Proposition~\ref{thm:coherent-family-has-uniq-central-char}. Namelly\dots 260 261 \begin{lemma}\label{thm:lemma6.1} 262 Let \(\beta \in \Sigma\) and \(\lambda \notin P^+\) be such that. 263 \(L(\lambda)\) is bounded and \(\lambda(H_\beta) \notin \mathbb{N}\). Then 264 \(L(\sigma_\beta \bullet \lambda) \subset \mExt(L(\lambda))\). In particular, 265 if \(\sigma_\beta \bullet \lambda \notin P^+\) then \(L(\sigma_\beta)\) is a 266 bounded infinite-dimensional \(\mathfrak{g}\)-module and 267 \(\mExt(L(\sigma_\beta \bullet \lambda)) \cong \mExt(L(\lambda))\). 268 \end{lemma} 269 270 \begin{note} 271 We should point out that, while it may very well be that \(\sigma_\beta 272 \bullet \lambda \in P^+\), there is generally only a slight chance of such an 273 event happening. Indeed, given \(\lambda \in \mathfrak{h}^*\), its orbit \(W 274 \bullet \lambda\) meets \(P^+\) precisely once, so that the probability of 275 \(\sigma_\beta \bullet \lambda \in P^+\) for some random \(\lambda \in 276 \mathfrak{h}^*\) is only \(\sfrac{1}{|W \bullet \lambda|}\). With the odds 277 stacked in our favor, we will later be able to exploit the second part of 278 Lemma~\ref{thm:lemma6.1} without much difficulty! 279 \end{note} 280 281 While technical in nature, this lemma already allows us to classify all 282 semisimple irreducible coherent \(\mathfrak{sl}_2(K)\)-families. 283 284 % TODO: Add a diagram of the locus of weights λ such that L(λ) is 285 % infinite-dimensional and bounded 286 \begin{example} 287 Let \(\mathfrak{g} = \mathfrak{sl}_2(K)\). It follows from 288 Example~\ref{ex:sl2-verma} that \(M(\lambda)\) is a bounded 289 \(\mathfrak{sl}_2(K)\) of degree \(1\), so that \(L(\lambda)\) is bounded -- 290 with \(\deg L(\lambda) = 1\) -- for all \(\lambda \in K \cong 291 \mathfrak{h}^*\). In addition, a simple calculation shows \(W \bullet 292 \lambda = \{\lambda, - \lambda - 2\}\). This implies that if \(\lambda, \mu 293 \notin P^+ = \mathbb{N}\) are such that \(\mExt(L(\lambda)) \cong 294 \mExt(L(\mu))\) then \(\mu = \lambda\) or \(\mu = - \lambda - 2\). Finally, 295 by Lemma~\ref{thm:lemma6.1} the converse also holds: if \(\lambda, - \lambda 296 - 2 \notin P^+\) then \(\mExt(L(\lambda)) \cong \mExt(L(- \lambda - 2))\). 297 \end{example} 298 299 % TODO: Add a transition here 300 301 \section{Coherent \(\mathfrak{sp}_{2n}(K)\)-families} 302 303 % TODO: Fix n >= 2 304 305 Consider the Cartan subalgebra \(\mathfrak{h} \subset \mathfrak{sp}_{2n}(K)\) 306 of diagonal matrices, as in Example~\ref{ex:cartan-of-sp}, and the basis 307 \(\Sigma = \{\beta_1, \ldots, \beta_n\}\) for \(\Delta\) given by \(\beta_i = 308 \epsilon_i - \epsilon_{i+1}\) for \(i < n\) and \(\beta_n = 2 \epsilon_n\). 309 Here \(\epsilon_i : \mathfrak{h} \to K\) is the linear functional which yields 310 the \(i\)-th entry of the diagonal of a given matrix, as described in 311 Example~\ref{ex:sp-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 + 312 \cdots + \sfrac{1}{2} \beta_n\). 313 314 % TODO: Add some comments on the proof of this: verifying that these conditions 315 % are necessary is a purely combinatorial affair, while checking that these are 316 % sufficient involves some analysis envolving the Shane-Weil module 317 \begin{lemma}\label{thm:sp-bounded-weights} 318 Then \(L(\lambda)\) is bounded if, and only if 319 \begin{enumerate} 320 \item \(\lambda(H_{\beta_i})\) is non-negative integer for all \(i \ne n\). 321 \item \(\lambda(H_{\beta_n}) \in \frac{1}{2} + \mathbb{Z}\). 322 \item \(\lambda(H_{\beta_{n - 1}} + 2 H_{\beta_n}) \ge -2\). 323 \end{enumerate} 324 \end{lemma} 325 326 % TODO: Note that we need a better set of parameters to the space of weights 327 % such that L(λ) is bounded 328 329 % TODO: Revise the notation for this? I don't really like calling this 330 % bijection m 331 \begin{proposition}\label{thm:better-sp(2n)-parameters} 332 The map 333 \begin{align*} 334 m : \mathfrak{h}^* & \to K^n \\ 335 \lambda & 336 \mapsto 337 ( 338 \kappa(\epsilon_1, \lambda+\rho), 339 \ldots, 340 \kappa(\epsilon_n, \lambda+\rho) 341 ) 342 \end{align*} 343 is \(W\)-equivariant bijection, where the action \(W \cong S_n \ltimes 344 (\mathbb{Z}/2\mathbb{Z})^n\) on \(\mathfrak{h}^*\) is given by the dot action 345 and the action of \(W\) on \(K^n\) is given my permuting coordinates and 346 multiplying them by \(\pm 1\). A weight \(\lambda \in \mathfrak{h}^*\) 347 satisfies the conditions of Lemma~\ref{thm:sp-bounded-weights} if, and 348 only if \(m(\lambda)_i \in \sfrac{1}{2} + \mathbb{Z}\) for all \(i\) and 349 \(m(\lambda)_1 > m(\lambda)_2 > \cdots > m(\lambda)_{n - 1} > \pm 350 m(\lambda)_n\). 351 \end{proposition} 352 353 \begin{proof} 354 The fact \(m : \mathfrak{h}^* \to K^n\) is a bijection is clear from the fact 355 that \(\{\epsilon_1, \ldots, \epsilon_n\}\) is an orthonormal basis for 356 \(\mathfrak{h}^*\). Veryfying that \(L(\lambda)\) is bounded if, and only if 357 \(m(\lambda)_1 > m(\lambda)_2 > \cdots > m(\lambda)_{n - 1} > \pm 358 m(\lambda)_n\) is also a simple combinatorial affair. 359 360 The only part of the statement worth proving is the fact that \(m\) is an 361 equivariant map, which is equivalent to showing the map 362 \begin{align*} 363 \mathfrak{h}^* & \to K^n \\ 364 \lambda & 365 \mapsto (\kappa(\epsilon_1, \lambda), \ldots, \kappa(\epsilon_n, \lambda)) 366 \end{align*} 367 is equivariant with respect to the natural action of \(W\) on 368 \(\mathfrak{h}^*\). But this also clear from the isomorphism \(W \cong S_n 369 \ltimes (\mathbb{Z}/2\mathbb{Z})^n\), as described in 370 Example~\ref{ex:sp-weyl-group}: \((\sigma_i, (\bar 0, \ldots, \bar 0)) = 371 \sigma_{\beta_i}\) permutes \(\epsilon_i\) and \(\epsilon_{i + 1}\) for \(i < 372 n\) and \((1, (\bar 0, \ldots, \bar 0, \bar 1)) = \sigma_{\beta_n}\) flips 373 the sign of \(\epsilon_n\). Hence \(m(\sigma_{\beta_i} \cdot \epsilon_j) = 374 \sigma_{\beta_i} \cdot m(\epsilon_j)\) for all \(i\) and \(j\). Since \(W\) 375 is generated by the \(\sigma_{\beta_i}\), this implies that the required map 376 is equivariant. 377 \end{proof} 378 379 \begin{definition} 380 We denote by \(\mathscr{B}\) the set of the \(m \in (\sfrac{1}{2} + 381 \mathbb{Z})^n\) such that \(m_1 > m_2 > \cdots > m_{n - 1} > \pm m_n\). We 382 also consider the canonical partition \(\mathscr{B} = \mathscr{B}^+ \cup 383 \mathscr{B}^-\) where \(\mathscr{B}^+ = \{ m \in \mathscr{B} : m_n > 0 \}\) 384 and \(\mathscr{B}^- = \{ m \in \mathscr{B} : m_n < 0\}\). 385 \end{definition} 386 387 \begin{theorem}[Mathieu] 388 Given \(\lambda\) and \(\mu\) satisfying the conditions of 389 Lemma~\ref{thm:sp-bounded-weights}, \(\mExt(L(\lambda)) \cong 390 \mExt(L(\mu))\) if, and only if \(m(\lambda)_i = m(\mu)_i\) for \(i < n\) and 391 \(m(\lambda)_n = \pm m(\mu)_n\). In particular, the isomorphism classes of 392 semisimple irreducible coherent \(\mathfrak{sp}_{2n}(K)\)-families are 393 parameterized by \(\mathscr{B}^+\). 394 \end{theorem} 395 396 \begin{proof} 397 Let \(\lambda, \mu \notin P^+\) be such that \(L(\lambda)\) and \(L(\mu)\), 398 so that \(m(\lambda), m(\mu) \in \mathscr{B}\). 399 400 Suppose \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\). By 401 Proposition~\ref{thm:coherent-family-has-uniq-central-char}, \(\chi_\lambda = 402 \chi_\mu\). It thus follows from the Harish-Chandra Theorem that \(\mu \in W 403 \bullet \lambda\). Since \(m\) is equivariant, \(m(\mu) \in W \cdot 404 m(\lambda)\). But the only elements in of \(\mathscr{B}\) in \(W \cdot 405 m(\lambda)\) are \(m(\lambda)\) and \((m(\lambda)_1, m(\lambda)_2, \ldots, 406 m(\lambda)_{n-1}, - m(\lambda)_n)\). 407 408 Conversely, if \(m(\lambda)_i = m(\mu)_i\) for \(i < n\) and \(m(\mu)_n = - 409 m(\lambda)_n\) then \(m(\mu) = \sigma_{\beta_n} \cdot m(\lambda)\) and \(\mu 410 = \sigma_n \bullet \lambda\). Since \(m(\lambda) \in \mathscr{B}\), 411 \(\lambda(H_{\beta_n}) \in \sfrac{1}{2} + \mathbb{Z}\) and thus 412 \(\lambda(H_{\beta_n}) \notin \mathbb{N}\). Hence by 413 Lemma~\ref{thm:lemma6.1} \(L(\mu) \subset \mExt(L(\lambda))\) and 414 \(\mExt(L(\mu)) \cong \mExt(L(\lambda))\). 415 416 For each semisimple irreducible coherent \(\mathfrak{sp}_{2n}(K)\)-family 417 \(\mathcal{M}\) there is some \(m = m(\lambda) \in \mathscr{B}\) such that 418 \(\mathcal{M} = \mExt(L(\lambda))\). The only other \(m' \in \mathscr{B}\) 419 which generates the same coherent family as \(m\) is \(m' = \sigma_{\beta_n} 420 \cdot m\). Since \(m\) and \(m'\) lie in different elements of the partition 421 \(\mathscr{B} = \mathscr{B}^+ \cup \mathscr{B}^-\), the is a unique \(m'' = 422 m(\nu) \in \mathscr{B}^+\) -- either \(m\) or \(m'\) -- such that 423 \(\mathcal{M} \cong \mExt(L(\nu))\). 424 \end{proof} 425 426 \section{Coherent \(\mathfrak{sl}_n(K)\)-families} 427 428 % TODO: Fix n >= 3 429 430 Consider the Cartan subalgebra \(\mathfrak{h} \subset \mathfrak{sl}_n(K)\) of 431 diagonal matrices, as in Example~\ref{ex:cartan-of-sl}, and the basis \(\Sigma 432 = \{\beta_1, \ldots, \beta_{n-1}\}\) for \(\Delta\) given by \(\beta_i = 433 \epsilon_i - \epsilon_{i+1}\) for \(i < n\). Here \(\epsilon_i : \mathfrak{h} 434 \to K\) is the linear functional which yields the \(i\)-th entry of the 435 diagonal of a given matrix, as described in 436 Example~\ref{ex:sl-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 + 437 \cdots + \sfrac{1}{2} \beta_{n - 1}\). 438 439 % TODO: Add some comments on the proof of this: while the proof that these 440 % conditions are necessary is a purely combinatorial affair, the proof of the 441 % fact that conditions (ii) and (iii) imply L(λ) is bounded requires some 442 % results on the connected components of of the graph 𝓑 (which we will only 443 % state later down the line) 444 \begin{lemma}\label{thm:sl-bounded-weights} 445 Let \(\lambda \notin P^+\) and \(A(\lambda) = \{ i : \lambda(H_{\beta_i})\ 446 \text{is \emph{not} a non-negative integer}\}\). Then \(L(\lambda)\) is 447 bounded if, and only if one of the following assertions holds. 448 \begin{enumerate} 449 \item \(A(\lambda) = \{1\}\) or \(A(\lambda) = \{n - 1\}\). 450 \item \(A(\lambda) = \{i\}\) for some \(1 < i < n - 1\) and \((\lambda + 451 \rho)(H_{\beta_{i - 1}} + H_{\beta_i})\) or \((\lambda + 452 \rho)(H_{\beta_i} + H_{\beta_{i + 1}})\) is a positive integer. 453 \item \(A(\lambda) = \{i, i + 1\}\) for some \(1 \le i < n - 1\) and 454 \((\lambda + \rho)(H_{\beta_i} + H_{\beta_{i + 1}})\) is a positive 455 integer. 456 \end{enumerate} 457 \end{lemma} 458 459 \begin{definition} 460 A \emph{\(\mathfrak{sl}_n\)-sequence} \(m\) is a \(n\)-tuple \(m = (m_1, 461 \ldots, m_n) \in K^n\) such that \(m_1 + \cdots + m_n = 0\). 462 \end{definition} 463 464 \begin{definition} 465 A \(k\)-tuple \(m = (m_1, \ldots, m_k) \in K^k\) is called \emph{ordered} if 466 \(m_i - m_{i + 1}\) is a positive integer for all \(i < k\). 467 \end{definition} 468 469 % TODO: Revise the notation for this? I don't really like calling this 470 % bijection m 471 \begin{proposition}\label{thm:better-sl(n)-parameters} 472 The map 473 \begin{align*} 474 m : \mathfrak{h}^* & 475 \to \{ \mathfrak{sl}_n\textrm{\normalfont-sequences} \} \\ 476 \lambda & 477 \mapsto 478 2n 479 ( 480 \kappa(\epsilon_1, \lambda + \rho), 481 \ldots, 482 \kappa(\epsilon_n, \lambda + \rho) 483 ) 484 \end{align*} 485 is \(W\)-equivariant bijection, where the action \(W \cong S_n\) on 486 \(\mathfrak{h}^*\) is given by the dot action and the action of \(W\) on the 487 space of \(\mathfrak{sl}_n\)-sequences is given my permuting coordinates. A 488 weight \(\lambda \in \mathfrak{h}^*\) satisfies the conditions of 489 Lemma~\ref{thm:sl-bounded-weights} if, and only if \(m(\lambda)\) is 490 \emph{not} ordered, but becomes ordered after removing one term. 491 \end{proposition} 492 493 % TODO: Note beforehand that κ(H_α, ⋅) is always a multiple of α. This is 494 % perhaps better explained when defining H_α 495 The proof of this result is very similar to that of 496 Proposition~\ref{thm:better-sp(2n)-parameters} in spirit: the equivariance of 497 the map \(m : \mathfrak{h}^* \to \{ \mathfrak{sl}_n\textrm{-sequences} \}\) 498 follows from the nature of the isomorphism \(W \cong S_n\) as described in 499 Example~\ref{ex:sl-weyl-group}, while the rest of the proof amounts to simple 500 technical verifications. The number \(2 n\) is a normalization constant chosen 501 because \(\lambda(H_\beta) = 2 n \, \kappa(\lambda, \beta)\) for all \(\lambda 502 \in \mathfrak{h}^*\) and \(\beta \in \Sigma\). Hence \(m(\lambda)\) is uniquely 503 characterized by the property that \((\lambda + \rho)(H_{\beta_i}) = 504 m(\lambda)_i - m(\lambda)_{i+1}\) for all \(i\), which is relevant to the proof 505 of the equivalence between the contiditions of 506 Lemma~\ref{thm:sl-bounded-weights} and those explained in the last statement of 507 Proposition~\ref{thm:better-sl(n)-parameters}. 508 509 % TODO: Explain the significance of 𝓑 + and 𝓑 -: these are the subsets whose 510 % union corresponds to condition (i) 511 \begin{definition} 512 We denote by \(\mathscr{B}\) the set of \(\mathfrak{sl}_n\)-sequences \(m\) 513 which are \emph{not} ordered, but becomes ordered after removing one term. We 514 also consider the \emph{extremal} subsets \(\mathscr{B}^+ = \{m \in 515 \mathscr{B} : (\widehat{m_1}, m_2, \ldots, m_n) \ \text{is ordered}\}\) and 516 \(\mathscr{B}^- = \{m \in \mathscr{B} : (m_1, \ldots, m_{n - 1}, 517 \widehat{m_n}) \ \text{is ordered}\}\). 518 \end{definition} 519 520 % TODO: Add a picture of parts of 𝓑 for n = 3 in here 521 522 % TODO: Explain that for each m ∈ 𝓑 there is a unique i such that so that 523 % m_i - m_i+1 is not a positive integer. For m ∈ 𝓑 + this is i = 1, while for 524 % m ∈ 𝓑 - this is i = n-1 525 The issue here is that the relationship between \(\lambda, \mu \in P^+\) with 526 \(m(\lambda), m(\mu) \in \mathscr{B}\) and \(\mExt(L(\lambda)) \cong 527 \mExt(L(\mu))\) is more complicated than in the case of \(\mathfrak{sp}_{2 528 n}(K)\). Nevertheless, Lemma~\ref{thm:lemma6.1} affords us a criteria for 529 verifying that \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\). For \(\sigma = 530 \sigma_i\) and the weight \(\lambda + \rho\), the hypothesis of 531 Lemma~\ref{thm:lemma6.1} translates to \(m(\lambda)_i - m(\lambda)_{i+1} = 532 (\lambda + \rho)(H_{\beta_i}) \notin \mathbb{N}\). If \(m(\lambda) \in 533 \mathscr{B}\), this is equivalent to requiring that \(m(\lambda)\) is not 534 ordered, but becomes ordered after removing its \(i\)-th term. This discussions 535 losely inspires the following definition, which endows the set \(\mathscr{B}\) 536 with the structure of a directed graph. 537 538 \begin{definition} 539 Given \(m, m' \in \mathscr{B}\), say there is an arrow \(m \to m'\) if there 540 some \(i\) such that \(m_i - m_{i + 1}\) is \emph{not} a positive integer and 541 \(m' = \sigma_i \cdot m\). 542 \end{definition} 543 544 It should then be obvious from Lemma~\ref{thm:lemma6.1} that\dots 545 546 \begin{proposition}\label{thm:arrow-implies-ext-eq} 547 Let \(\lambda \notin P^+\) be such that \(L(\lambda)\) is bounded -- so that 548 \(m(\lambda) \in \mathscr{B}\) -- and suppose that \(\mu \in \mathfrak{h}^*\) 549 is such that \(m(\mu) \in \mathscr{B}\) and there is an arrow \(m(\lambda) 550 \to m(\mu)\). Then \(L(\mu)\) is also bounded and \(\mExt(L(\mu)) \cong 551 \mExt(L(\lambda))\). 552 \end{proposition} 553 554 A weight \(\lambda \in \mathfrak{h}^*\) is called \emph{regular} if \((\lambda 555 + \rho)(H_\alpha) \ne 0\) for all \(\alpha \in \Delta\). In terms of 556 \(\mathfrak{sl}_n\)-sequences, \(\lambda\) is regular if, and only if 557 \(m(\lambda)_i \ne m(\lambda)_j\) for all \(i \ne j\). It thus makes sence to 558 call a \(\mathfrak{sl}_n\)-sequence regular or singular if \(m_i \ne m_j\) for 559 all \(i \ne j\) or \(m_i = m_j\) for some \(i \ne j\), respectively. Similarly, 560 \(\lambda\) is integral if, and only if \(m(\lambda)_i - m(\lambda)_j \in 561 \mathbb{Z}\) for all \(i\) and \(j\), so it makes sence to call a 562 \(\mathfrak{sl}_n\)-sequence \(m\) integral if \(m_i - m_j \in \mathbb{Z}\) for 563 all \(i\) and \(j\). 564 565 % TODO: Restate the notation for σ_i beforehand 566 \begin{lemma}\label{thm:connected-components-B-graph} 567 The connected component of some \(m \in \mathscr{B}\) is given by the 568 following. 569 \begin{enumerate} 570 \item If \(m\) is regular and integral then there exists\footnote{Notice 571 that in this case $m' \notin \mathscr{B}$, however.} a unique ordered 572 \(m' \in W \cdot m\), in which case the connected component of \(m\) is 573 given by 574 \[ 575 \begin{tikzcd}[cramped, row sep=small] 576 \sigma_1 \sigma_2 \cdots \sigma_i \cdot m' \rar & 577 \sigma_2 \cdots \sigma_i \cdot m' \rar & 578 \cdots \rar & 579 \sigma_{i-1} \sigma_i \cdot m' 580 \ar[rounded corners, 581 to path={ -- ([xshift=4ex]\tikztostart.east) 582 |- (X.center) \tikztonodes 583 -| ([xshift=-4ex]\tikztotarget.west) 584 -- (\tikztotarget)}]{dlll}[at end]{} \\ 585 \sigma_i \cdot m' & 586 \sigma_{i+1} \sigma_i \cdot m' \lar & 587 \cdots \lar & 588 \sigma_{n-1} \cdots \sigma_i \cdot m' \lar & 589 \end{tikzcd} 590 \] 591 for some unique \(i\), with \(\sigma_1 \cdots \sigma_i \cdot m' \in 592 \mathscr{B}^+\) and \(\sigma_{n-1} \cdots \sigma_i \cdot m' \in 593 \mathscr{B}^-\). 594 595 % TODOOO: What happens when i = 1?? Do we need to suppose i > 1? 596 % TODO: For instance, consider m = (1, 1, -2) 597 \item If \(m\) is singular then there exists unique \(m' \in W \cdot m\) 598 and \(i\) such that \(m'_i = m'_{i + 1}\) and \((m_1', \cdots, m_{i-1}', 599 \widehat{m_i'}, m_{i + 1}', \ldots, m_n')\) is ordered, in which 600 case the 601 connected component of \(m\) is given by 602 \[ 603 \begin{tikzcd}[cramped, row sep=small] 604 \sigma_1 \sigma_2 \cdots \sigma_{i-1} \cdot m' \rar & 605 \sigma_2 \cdots \sigma_{i-1} \cdot m' \rar & 606 \cdots \rar & 607 \sigma_{i-1} \cdot m' 608 \ar[rounded corners, 609 to path={ -- ([xshift=4ex]\tikztostart.east) 610 |- (X.center) \tikztonodes 611 -| ([xshift=-4ex]\tikztotarget.west) 612 -- (\tikztotarget)}]{dlll}[at end]{} \\ 613 m' & 614 \sigma_{i+1} \cdot m' \lar & 615 \cdots \lar & 616 \sigma_{n-1} \cdots \sigma_{i+1} \cdot m' \lar & 617 \end{tikzcd} 618 \] 619 with \(\sigma_1 \cdots \sigma_{i-1} \cdot m' \in \mathscr{B}^+\) and 620 \(\sigma_{n-1} \cdots \sigma_{i+1} \cdot m' \in \mathscr{B}^-\). 621 622 \item If \(m\) is non-integral then there exists unique \(m' \in W \cdot 623 m\) with \(m' \in \mathscr{B}^+\), in which case the connected component 624 of \(m\) is given by 625 \[ 626 \begin{tikzcd}[cramped] 627 m' \rar & 628 \sigma_1 \cdot m' \rar \lar & 629 \sigma_2 \sigma_1 \cdot m' \rar \lar & 630 \cdots \rar \lar & 631 \sigma_{n-1} \cdots \sigma_1 \cdot m' \lar & 632 \end{tikzcd} 633 \] 634 with \(\sigma_{n-1} \cdots \sigma_1 \cdot m' \in \mathscr{B}^-\). 635 \end{enumerate} 636 \end{lemma} 637 638 % TODO: Notice that this gives us that if m(λ)∈ 𝓑 then L(λ) is bounded: for λ 639 % ∈ 𝓑 + ∪ 𝓑 - we stablish this by hand, and for the general case it suffices to 640 % notice that there is always some path μ → ... → λ with μ ∈ 𝓑 + ∪ 𝓑 - 641 642 \begin{theorem}[Mathieu] 643 Given \(\lambda, \mu \notin P^+\) with \(L(\lambda)\) and \(L(\mu)\) bounded, 644 \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\) if, and only if \(m(\lambda)\) and 645 \(m(\mu)\) lie in the same connected component of \(\mathscr{B}\). In 646 particular, the isomorphism classes of semisimple irreducible coherent 647 \(\mathfrak{sl}_n(K)\)-families are parameterized by the set 648 \(\pi_0(\mathscr{B})\) of the connected components of \(\mathscr{B}\), as 649 well as by \(\mathscr{B}^+\). 650 \end{theorem} 651 652 \begin{proof} 653 Let \(\lambda, \mu \notin P^+\) be such that \(L(\lambda)\) and \(L(\mu)\), 654 so that \(m(\lambda), m(\mu) \in \mathscr{B}\). 655 656 It is clear from Proposition~\ref{thm:arrow-implies-ext-eq} that if 657 \(m(\lambda)\) and \(m(\mu)\) lie in the same connected component of 658 \(\mathscr{B}\) then \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\). On the other 659 hand, if \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\) then \(\chi_\lambda = 660 \chi_\mu\) and thus \(\mu \in W \bullet \lambda\). We now investigate which 661 elements of \(W \bullet \lambda\) satisfy the conditions of 662 Lemma~\ref{thm:sl-bounded-weights}. To do so, we describe the set 663 \(\mathscr{B} \cap W \cdot m(\lambda)\). 664 665 % Great migué 666 If \(\lambda\) is regular and integral then the only permutations of 667 \(m(\lambda)\) which lie in \(\mathscr{B}\) are \(\sigma_k \sigma_{k+1} 668 \cdots \sigma_i \cdot m'\) for \(k \le i\) and \(\sigma_k \sigma_{k-1} \cdots 669 \sigma_i \cdot m'\) for \(k \ge i\), where \(m'\) is the unique ordered 670 element of \(W \cdot m(\lambda)\). Hence by 671 Lemma~\ref{thm:connected-components-B-graph} \(\mathscr{B} \cap W \cdot 672 m(\lambda)\) is the union of the connected components of the \(\sigma_i \cdot 673 m'\) for \(i \le n\). On the other hand, if \(\lambda\) is singular or 674 non-integral then the only permutations of \(m(\lambda)\) which lie in 675 \(\mathscr{B}\) are the ones from the connected component of \(m(\lambda)\) 676 in \(\mathscr{B}\), so that \(\mathscr{B} \cap W \cdot m(\lambda)\) is 677 exactly the connected component of \(m(\lambda)\). 678 679 In both cases, we can see that if \(B(\lambda)\) is the set of the \(m' = 680 m(\mu) \in \mathscr{B}\) such that \(\mExt(L(\mu)) \cong \mExt(L(\lambda))\) 681 then \(B(\lambda) \subset \mathscr{B} \cap W \cdot m(\lambda)\) is contain in 682 a union of connected components of \(\mathscr{B}\) -- including that of 683 \(m(\lambda)\) itself. We now claim that \(B(\lambda)\) is exactly the 684 connected component of \(m(\lambda)\). This is already clear when \(\lambda\) 685 is singular or non-integral, so we may assume that \(\lambda\) is regular and 686 integral, in which case every other \(\mu \in W \bullet \lambda\) is regular 687 and integral. 688 689 % Great migué 690 In this situation, \(m(\mu) \in \mathscr{B}^+\) implies \(\mu(H_{\beta_1}) = 691 m(\mu)_1 - m(\mu)_2 \in \mathbb{Z}\) is negative. But it follows from 692 Lemma~\ref{thm:lemma6.1} that for each \(\beta \in \Sigma\) there is at 693 most one \(\mu \notin P^+\) with \(\mExt(L(\mu)) \cong \mExt(L(\lambda))\) 694 such that \(\mu(H_\beta)\) is a negative integer -- see Lemma~6.5 of 695 \cite{mathieu}. Hence there is at most one \(m' \in \mathscr{B}^+ \cap W 696 \cdot m(\lambda)\). Since every connected component of \(\mathscr{B}\) meets 697 \(\mathscr{B}^+\) -- see Lemma~\ref{thm:connected-components-B-graph} -- this 698 implies \(B(\lambda)\) is precisely the connected component of 699 \(m(\lambda)\). 700 701 Another way of putting it is to say that \(\mExt(L(\lambda)) \cong 702 \mExt(L(\mu))\) if, and only if \(m(\lambda)\) and \(m(\mu)\) lie in the same 703 connected component -- which is, of course, precisely the first part of our 704 theorem! There is thus a one-to-one correspondance between 705 \(\pi_0(\mathscr{B})\) and the isomorphism classes of semisimple irreducible 706 coherent \(\mathfrak{sl}_n(K)\)-families. Since every connected component of 707 \(\mathscr{B}\) meets \(\mathscr{B}^+\) precisely once -- again, see 708 Lemma~\ref{thm:connected-components-B-graph} -- we also get that such 709 isomorphism classes are parameterized by \(\mathscr{B}^+\). 710 \end{proof} 711 712 % TODO: Change this 713 % I don't really think these notes bring us to this conclusion 714 % If anything, these notes really illustrate the incredible vastness of the 715 % ocean of representation theory, how unknowable it is, and the remarkable 716 % amount of engenuity required to explore it 717 This construction also brings us full circle to the beginning of these notes, 718 where we saw in Proposition~\ref{thm:geometric-realization-of-uni-env} that 719 \(\mathfrak{g}\)-modules may be understood as geometric objects. In fact, 720 throughout the previous four chapters we have seen a tremendous number of 721 geometrically motivated examples, which further emphasizes the connection 722 between representation theory and geometry. I would personally go as far as 723 saying that the beautiful interplay between the algebraic and the geometric is 724 precisely what makes representation theory such a fascinating and charming 725 subject. 726 727 Alas, our journey has come to an end. All it is left is to wonder at the beauty 728 of Lie algebras and their representations. 729 730 \label{end-47}