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}