Proof. Suppose the good filtration $F_\bullet M$ satisfies the condition of finiteness. Then, as the action of $\mathbf {S}$ on M preserves each degree $F_{k} M$ , the action of $\mathbf {S}$ on M is locally finite; hence, $M\in \mathrm {O}(\mathbf {H})$ . Conversely, suppose that $M\in \mathrm {O}(\mathbf {H})$ and $\left \{F_k M \right \}_{k\in \mathbb {Z}}$ is a good filtration with respect to the length filtration $F^{\mathrm {lg}}_\bullet \mathbf {H}$ . Since ${\mathrm {gr}}^F M$ is finitely generated over ${\mathrm {gr}}^{\mathrm {lg}}\mathbf {H}$ , we can find a finite-dimensional generating graded subspace $V\subset {\mathrm {gr}}^F M$ . Hence, we have

$$\begin{align*}{\mathrm{gr}}^F_k M = \bigoplus_{j\in \mathbb{Z}} {\mathrm{gr}}^{\mathrm{lg}}_j \mathbf{H}(V\cap {\mathrm{gr}}^F_{k-j} M) = \bigoplus_{j\in \mathbb{N}}\bigoplus_{\ell(w) = j}[w]\mathbf{S} (V\cap {\mathrm{gr}}^F_{k-j} M). \end{align*}$$

By the hypothesis that $M\in \mathrm {O}(\mathbf {H})$ , Corollary 1.19 implies that the space $\mathbf {S} (V\cap {\mathrm {gr}}^F_{k-j} M)$ is finite-dimensional. Since ${\mathrm {gr}}^F_{k-j} M$ vanishes for $j \gg 0$ by the finite generation, it follows that $\dim {\mathrm {gr}}^F_k M < \infty $ and, consequently, $\dim F_k M < \infty $ .

Proof. See [Reference Losev11, p. 3.4.3].

Proof. It is obvious that the induced filtration of a HC filtration on a sub-bimodule or a quotient bimodule is again a HC filtration; thus, $\mathrm {HC}(\mathbf {H})$ is closed under formation of sub-objects and quotient objects. Suppose that M is an $\mathbf {H}$ -bimodule, $M'\subset M$ is a sub- $\mathbf {H}$ -bimodule and $M" = M / M'$ such that $M'$ and $M"$ are in $\mathrm {HC}(\mathbf {H})$ . Pick a good filtration $\left \{ F_k M \right \}_{k\in \mathbb {Z}}$ for M and let $\left \{ F_k M' \right \}_{k\in \mathbb {Z}}$ and $\left \{ F_k M" \right \}_{k\in \mathbb {Z}}$ be the induced filtrations. We have a short exact sequence

(2.4) $$ \begin{align} 0\to {\mathrm{gr}}^F M'\to {\mathrm{gr}}^F M\to {\mathrm{gr}}^F M"\to 0. \end{align} $$

Since $M'$ and $M"$ are HC and since $\left \{ F_k M' \right \}_{k\in \mathbb {Z}}$ and $\left \{ F_k M" \right \}_{k\in \mathbb {Z}}$ are good (using (2.4)), it follows that they are also HC filtrations. We see that the filtration $\{F_k M\}$ is also HC by using (2.4) again.

Proof. Let $\{ F_k M \}_{k\in \mathbb {Z}}$ and $\{ F_k N \}_{k\in \mathbb {Z}}$ be HC filtrations on M and N, respectively. Consider the Rees bimodules

$$\begin{align*}M_{\delta} = \bigoplus_{k\in \mathbb{Z}} \delta^k F_k M,\quad N_{\delta} = \bigoplus_{k\in \mathbb{Z}} \delta^k F_k N. \end{align*}$$

They are finitely generated graded $\mathbf {H}_{\delta }$ -bimodules flat over $\mathbb {C}[\delta ]$ . The extension group ${\mathrm {Ext}}^k_{\mathbf {H}_{\delta }}(M_{\delta }, N_{\delta })$ is a finitely generated graded $\mathbf {H}_{\delta }$ -bimodule. We have an exact sequence

$$\begin{align*}{\mathrm{Ext}}^k_{\mathbf{H}_{\delta}}(M_{\delta}, N_{\delta})\xrightarrow{\delta\cdot}{\mathrm{Ext}}^k_{\mathbf{H}_{\delta}}(M_{\delta}, N_{\delta})\to {\mathrm{Ext}}^k_{\mathbf{H}_{\delta}}(M_\delta, N_{\delta}/(\delta)). \end{align*}$$

By the flatness of $M_{\delta }$ over $\mathbb {C}[\delta ]$ , there is an isomorphism:

$$\begin{align*}{\mathrm{Ext}}^k_{\mathbf{H}_{\delta}}(M_\delta, N_{\delta}/(\delta)) \cong {\mathrm{Ext}}^k_{\mathbf{H}_{\delta}/(\delta)}\left( M_{\delta}/(\delta), N_{\delta}/(\delta)\right)= {\mathrm{Ext}}^k_{{\mathrm{gr}}^{\mathrm{can}}\mathbf{H}}({\mathrm{gr}}^F M, {\mathrm{gr}}^F N). \end{align*}$$

We obtain an embedding

$$\begin{align*}{\mathrm{Ext}}^k_{\mathbf{H}_{\delta}}(M_{\delta}, N_{\delta}) / (\delta) \hookrightarrow {\mathrm{Ext}}^k_{{\mathrm{gr}}^{\mathrm{can}}\mathbf{H}}({\mathrm{gr}}^F M, {\mathrm{gr}}^F N). \end{align*}$$

The $\mathbb {Z}$ -grading on ${\mathrm {Ext}}^k_{\mathbf {H}_{\delta }}(M_{\delta }, N_{\delta })$ induces a filtration on the quotient

$$\begin{align*}{\mathrm{Ext}}^k_{\mathbf{H}_{\delta}}(M_{\delta}, N_{\delta}) / (\delta - 1) \cong {\mathrm{Ext}}^k_{\mathbf{H}}(M, N), \end{align*}$$

denoted by F. It is easy to show that the associated graded ${\mathrm {gr}}^F{\mathrm {Ext}}^k_{\mathbf {H}}(M, N)$ is a quotient of ${\mathrm {Ext}}^k_{\mathbf {H}_{\delta }}(M_{\delta }, N_{\delta })/(\delta )$ . By the assumption, for every $z\in Z({\mathrm {gr}}^{\mathrm {can}}\mathbf {H})$ , the operator $({\mathrm {ad}} z)^d$ vanishes on ${\mathrm {gr}}^F M$ and ${\mathrm {gr}}^F N$ when $d \gg 0$ ; hence, it is also the case on ${\mathrm {Ext}}^k_{{\mathrm {gr}}^{\mathrm {can}}\mathbf {H}}({\mathrm {gr}}^F M, {\mathrm {gr}}^F N)$ and on the subquotient bimodule ${\mathrm {gr}}^F{\mathrm {Ext}}^k_{\mathbf {H}}(M, N)$ . It follows that F is a HC filtration on ${\mathrm {Ext}}^k_{\mathbf {H}}(M, N)$ and the latter is a Harish-Chandra $\mathbf {H}$ -bimodule. The proof for ${\mathrm {Tor}}^{\mathbf {H}}_k(M, N)$ is similar.

Proof. It results immediately from Proposition 1.4 and Proposition 2.5.

Proof. We prove the statements for and leave those for to the reader. By Corollary 2.6, we have ${\mathrm {RHom}}_{\mathbf {H}}(B, \mathbf {H})\in \mathrm {D}^{\mathrm {b}}_{\mathrm {HC}}(\mathbf {H})$ . Since HC bimodules are finitely generated left $\mathbf {H}$ -modules, we have ${\mathrm {RHom}}_{\mathbf {H}}(B, \mathbf {H})\in \mathrm {D}^{\mathrm {b}}_{\mathrm {perf}}(\mathbf {H})$ when the right $\mathbf {H}$ -module structure is forgotten. Since $\mathbf {H}$ generates $\mathrm {D}^{\mathrm {b}}_{\mathrm {perf}}(\mathbf {H})$ as thick subcategory, we see that ${\mathrm {RHom}}_{\mathbf {H}}(B, M)\in \mathrm {D}^{\mathrm {b}}_{\mathrm {perf}}(\mathbf {H})$ for each $M\in \mathrm {D}^{\mathrm {b}}_{\mathrm {perf}}(\mathbf {H})$ .

Let $M\in \mathrm {O}(\mathbf {H})$ . We pick a good filtration $\left \{ F_k M \right \}_{k\in \mathbb {Z}}$ for M and, by Lemma 2.8 below, an excellent filtration $\left \{ F_k B \right \}_{k\in \mathbb {Z}}$ for B with respect to the length filtration $F^{\mathrm {lg}}_\bullet \mathbf {H}$ introduced in § 1.8. Set

$$\begin{align*}\mathbf{H}_{\zeta} = \bigoplus_{k\in \mathbb{N}}\zeta^k F^{\mathrm{lg}}_k \mathbf{H},\quad B_{\zeta} = \bigoplus_{k\in \mathbf{Z}}\zeta^k F_k B,\quad M_{\zeta} = \bigoplus_{k\in \mathbf{Z}}\zeta^k F_k M. \end{align*}$$

By Proposition 1.25, $F_k M$ is a finite-dimensional $\mathbf {S}$ -module for each $k\in \mathbb {Z}$ . Moreover, $B_{\zeta }$ is a $(\mathbf {H}_{\zeta }, \mathbf {H}_{\zeta })$ -bimodule and finite as left $\mathbf {H}_{\zeta }$ -module. By taking a resolution of $B_{\zeta }$ by finite graded-free left $\mathbf {H}_{\zeta }$ -modules, we deduce easily that ${\mathrm {Ext}}^k_{\mathbf {H}_{\zeta }}(B_{\zeta }, M_{\zeta })$ is a graded vector space and finite-dimensional in each degree and for each $k\in \mathbb {Z}$ ; moreover, it has the structure of graded left $\mathbf {H}_{\zeta }$ -module coming from the right $\mathbf {H}_{\zeta }$ -module structure on $B_{\zeta }$ . By the finite dimensionality, $\mathbf {S}$ acts locally finitely on ${\mathrm {Ext}}^k_{\mathbf {H}_{\zeta }}(B_{\zeta }, M_{\zeta })$ ; this implies that the $\mathbf {S}$ -action on the quotient ${\mathrm {Ext}}^k_{\mathbf {H}_{\zeta }}(B_{\zeta }, M_{\zeta }) / (\zeta - 1) \cong {\mathrm {Ext}}^k_{\mathbf {H}}(B, M)$ is also locally finite. It follows that ${\mathrm {Ext}}^k_{\mathbf {H}}(B, M)\in \mathrm {O}(\mathbf {H})$ for each $k\in \mathbb {Z}$ .

Proof. The proof is modeled on that of [Reference Losev11, p. 5.4.3]. We first show that for each finite left $\mathbf {S}$ -submodule $M\subseteq B$ , the product $N = M\mathbf {S}$ remains finite as left $\mathbf {S}$ -module. Let $\left \{ F^{\prime }_k B \right \}_{k\in \mathbf {Z}}$ be a HC filtration (Definition 2.1) on B and set $F^{\prime }_k M = M \cap F^{\prime }_k B$ , $F^{\prime }_k N = N \cap F^{\prime }_k B$ . Then ${\mathrm {gr}}^{F'} N = ({\mathrm {gr}}^{F'} M)({\mathrm {gr}}^{\mathrm {can}} \mathbf {S})$ holds, where ${\mathrm {gr}}^{\mathrm {can}}\mathbf {S}$ is the associated graded of restriction of the canonical filtration (§ 1.2.3) to $\mathbf {S}$ . Since the adjoint action of $({\mathrm {gr}}^{\mathrm {can}} \mathbf {S})^ W $ on ${\mathrm {gr}}^{F'} B$ is nilpotent and since ${\mathrm {gr}}^{\mathrm {can}} \mathbf {S}$ is finite over $({\mathrm {gr}}^{\mathrm {can}} \mathbf {S})^ W $ , it follows that ${\mathrm {gr}}^{F'} N$ is also finite over ${\mathrm {gr}}^{\mathrm {can}} \mathbf {S}$ . Hence, N is finite over $\mathbf {S}$ .

Recall the length filtration $F^{\mathrm {lg}}_{\bullet }\mathbf {H}$ from § 1.8. Now, let $V\subseteq B$ be a finite-dimensional generating subspace as left $\mathbf {H}$ -module. Let $\left \{ \overline b_j \right \}_{j = 1}^r$ be a spanning set for ${\mathrm {gr}}^{\mathrm {lg}}\mathbb {C} \hat {W}$ as left ${\mathrm {gr}}^{\mathrm {lg}}(\mathbb {C} Q^{\vee })^{ W }$ -module. We assume $\overline b_j$ is homogeneous of degree $d^{\prime }_j> 0$ except that $\overline b_1 = 1$ , and we choose a lifting $b_j$ of $\overline b_j$ in $(\mathbb {C} Q^{\vee })^ W $ . Let $\left \{ \overline z_k \right \}_{k=1}^s$ be a set of homogeneous $\mathbb {C}$ -algebra generators for ${\mathrm {gr}}^{\mathrm {lg}} (\mathbb {C} Q^{\vee })^ W $ of degree $\deg \overline z_k = d_k$ , and we choose a lifting $z_k$ of $\overline z_k$ in $(\mathbb {C} Q^{\vee })^ W $ . Define

$$\begin{align*}F_n B = \sum_{j=1}^r\sum_{\substack{n_0, n_1, \ldots, n_s\in \mathbb{N} \\ n_0 + d_1n_1 + \cdots + d_sn_s + d^{\prime}_j \le n}} (F^{\mathrm{lg}}_{n_0}\mathbf{H})\left(({\mathrm{ad}} z_s)^{n_s}\cdots ({\mathrm{ad}} z_1)^{n_1}V\right)b_j\mathbf{S} \end{align*}$$

Let’s show that ${\mathrm {gr}}^F B$ is finitely generated as left ${\mathrm {gr}}^{\mathrm {lg}} \mathbf {H}$ -module. Since the sum

$$\begin{align*}V' = \sum_{j=1}^r\sum_{n_1, \ldots, n_s\in \mathbb{N}} \mathbf{S}\left(({\mathrm{ad}} z_s)^{n_s}\cdots ({\mathrm{ad}} z_1)^{n_1}V\right)b_j\mathbf{S} \end{align*}$$

is a finite left $\mathbf {S}$ -module by the first paragraph, we can find a finite subset of homogeneous polynomials $I\subset \mathbf {S}$ such that the finite-dimensional subspace

$$\begin{align*}V" = \sum_{f\in I}\sum_{j=1}^r\sum_{n_1, \ldots, n_s\in \mathbb{N}} \mathbb{C} \left(({\mathrm{ad}} z_s)^{n_s}\cdots ({\mathrm{ad}} z_1)^{n_1}V\right)b_jf\subseteq V' \end{align*}$$

spans $V'$ as left $\mathbf {S}$ -module. Let $F_nV" = V"\cap F_nB$ ; then ${\mathrm {gr}}^F V"$ clearly generates ${\mathrm {gr}}^FB$ as left ${\mathrm {gr}}^{\mathrm {lg}}\mathbf {H}$ -module. Thus, ${\mathrm {gr}}^F B$ is a finite left ${\mathrm {gr}}^{\mathrm {lg}}\mathbf {H}$ -module.

Let’s show that $(F_k B)(F^{\mathrm {lg}}_l \mathbf {H})\subseteq (F_{k+l} B)$ . Clearly, it is enough to show that for $h\in F_k\mathbf {H}$ , $j \in \{ 1, \ldots , r\}$ and $v_0\in V$ , we have

$$\begin{align*}(({\mathrm{ad}} z_s)^{n_s} \cdots ({\mathrm{ad}} z_1)^{n_1}v_0)b_j h \in F_{k+n} B,\quad \text{where }n = n_1d_1 + \cdots + n_s d_s + d^{\prime}_j. \end{align*}$$

We prove this statement by induction on n. When $n = 0$ , we must have $d^{\prime }_j = 0$ , so $b_j = b_1 = 1$ . In other words, we need to prove that $v_0 h \in F_k B$ . We may assume that h is of the form $z b_{j'}$ , where $z = z_1^{m_1}\cdots z_s^{m_s}$ is a monomial and $j'\in \left \{ 1, \ldots ,r \right \}$ such that $m_1d_1 + \cdots + m_sd_s + d^{\prime }_j \le k$ . One can easily prove the following formula by induction:

$$\begin{align*}v_0h = \sum_{\substack{0\le m^{\prime}_1\le m_1 \\ \cdots \\ 0\le m^{\prime}_s\le m_s}}(-1)^{m^{\prime}_1 + \cdots + m^{\prime}_s}\binom{m_1}{m^{\prime}_1}\cdots\binom{m_s}{m^{\prime}_s} z_1^{m_1 - m^{\prime}_1} \cdots z_s^{m_s - m^{\prime}_s}\left(({\mathrm{ad}} z_s)^{m^{\prime}_s}\cdots({\mathrm{ad}} z_1)^{m^{\prime}_1}v_0\right)b_{j'}. \end{align*}$$

Since $z_1^{m_1 - m^{\prime }_1} \cdots z_s^{m_s - m^{\prime }_s} \in F_{d_1(m_1 - m^{\prime }_1) + \cdots + d_s(m_s - m^{\prime }_s)} \mathbf {H}$ , we deduce $v_0h\in F_k B$ . This is the initial step of the induction. Suppose $n> 0$ and the statement has been proven for smaller n. If $j\neq 1$ , we have $d^{\prime }_j> 0$ ; the induction hypothesis applied to $b_jh$ in place of h yields $(({\mathrm {ad}} z_s)^{n_s} \cdots ({\mathrm {ad}} z_1)^{n_1}v_0)(b_jh)\in F_{k+n} B$ . Suppose now that $j = 1$ . Let $i_0 = \max \left \{ i\;;\; n_i> 0 \right \}$ . Then

$$ \begin{align*} (({\mathrm{ad}} z_{i_0})^{n_{i_0}} \cdots ({\mathrm{ad}} z_1)^{n_1}v_0)h =\ & z_{i_0}(({\mathrm{ad}} z_{i_0})^{n_{i_0}-1} \cdots ({\mathrm{ad}} z_1)^{n_1}v_0) h \\ &- (({\mathrm{ad}} z_{i_0})^{n_{i_0}-1} \cdots ({\mathrm{ad}} z_1)^{n_1}v_0)z_{i_0} h. \end{align*} $$

Then, the induction hypothesis yields the result. It follows that $F_\bullet B$ is an $\mathbf {H}$ -bimodule filtration with respect to $F^{\mathrm {lg}}_\bullet \mathbf {H}$ and is good with respect to the left $\mathbf {H}$ -action. Analogously, we can construct an $\mathbf {H}$ -bimodule filtration $F^{\prime }_\bullet B$ with respect to $F^{\mathrm {lg}}_\bullet \mathbf {H}$ and that is good with respect to the right $\mathbf {H}$ -action. Since both filtrations are good over $\mathbf {H} \otimes \mathbf {H}^{\mathrm {op}}$ with respect to the filtration $F_{\bullet }^{\zeta }\otimes F_{\bullet }^{\zeta }$ , there exists $a \gg 0$ such that $F^{\prime }_{n-a} B \subseteq F_n B \subseteq F^{\prime }_{n+a}B$ for every $n\in \mathbb {Z}$ ; see [Reference Hotta, Takeuchi and Tanisaki7, p. D.1.3]. It follows that $F_\bullet B$ is also good for the right $\mathbf {H}$ -action, and thus excellent.

Proof. The proof, which we leave to the reader, is an easy exercise of commutative algebra involving the Chinese remainder theorem.

Proof. The left $\mathscr {H}$ -module structure on $\mathscr {C} B$ is described in the previous paragraph. The right $\mathscr {H}$ -module structure is defined as follows:

$$ \begin{align*} &\mathscr{C} B\otimes_{\mathbf{S}}\mathscr{H} = \mathscr{S}\otimes_{\mathbf{S}}B\otimes_{\mathbf{S}}\mathscr{S}\otimes_{\mathbf{S}}\mathbf{H} \\ \xrightarrow{{\mathrm{id}}_{\mathscr{S}}\otimes \eta_B\otimes {\mathrm{id}}_{\mathbf{H}}}&\mathscr{S}\otimes_{\mathbf{S}}\mathscr{S}\otimes_{\mathbf{S}}B\otimes_{\mathbf{S}}\mathbf{H} \xrightarrow{\mu_{\mathscr{S}}\otimes \rho_B} \mathscr{S}\otimes_{\mathbf{S}}B = \mathscr{C} B, \end{align*} $$

where $\rho _B$ is the right $\mathbf {H}$ -action on B. It is easy to verify that the left and right actions on B commute.

Proof. For $M\in \mathbf {H}\text{-}{\mathrm{Mod}}$ , we have isomorphisms

$$ \begin{align*} &\mathscr{C}(B)\otimes_{\mathscr{H}}\mathscr{C}(M) \cong B\otimes_{\mathbf{H}}\mathscr{H}\otimes_{\mathscr{H}}\mathscr{H}\otimes_{\mathbf{H}}M \\ &\cong B\otimes_{\mathbf{H}}\mathscr{H}\otimes_{\mathbf{H}}M\cong \mathscr{H}\otimes_{\mathbf{H}}B\otimes_{\mathbf{H}}M = \mathscr{C}(B\otimes_{\mathbf{H}}M). \end{align*} $$

They are bi-functorial in B and M. Taking the derived functors, we obtain the quasi-isomorphism:

(2.12) $$ \begin{align} \mathscr{C}(B\otimes^{\mathrm{L}}_{\mathbf{H}} M)\xrightarrow{\sim} \mathscr{C}(B)\otimes^{\mathrm{L}}_{\mathscr{H}}\mathscr{C}(M). \end{align} $$

This improves (ii) Given any $\mathscr {M}\in \mathscr {H}{\mathrm {-mod}}_{\mathrm {fl}}$ , we have $\mathscr {M}\cong \mathscr {C}(M)$ for some $M\in \mathrm {O}(\mathbf {H})$ by Theorem 1.16. Hence, (2.12) yields $\mathscr {C}(B)\otimes ^{\mathrm {L}}_{\mathscr {H}}\mathscr {M} \cong \mathscr {C}(B\otimes ^{\mathrm {L}}_{\mathbf {H}} M)$ . By Proposition 2.7, $B\otimes ^{\mathrm {L}}_{\mathbf {H}} M$ lies in $\mathrm {D}^{\mathrm {b}}_{\mathrm {O}}(\mathbf {H})$ , so $\mathscr {C}(B\otimes ^{\mathrm {L}}_{\mathbf {H}} M)$ is in $\mathrm {D}^{\mathrm {b}}_{\mathrm {fl}}(\mathscr {H})$ by Theorem 1.16 again. Therefore, the functor preserves $\mathrm {D}^{\mathrm {b}}_{\mathrm {fl}}(\mathscr {H})$ .

Given any $(c, \lambda )\in \mathfrak {P}\times \mathfrak {h}^{1}$ , the projectivity of $\mathscr {H} 1_{c, \lambda }$ yields $\mathscr {C}(B)\otimes ^{\mathrm {L}}_{\mathscr {H}}\mathscr {H} 1_{c, \lambda } \cong \mathscr {C}(B)1_{c, \lambda }$ . Let $\left \{ b_i \right \}_{i\in I}$ be a finite generating set of B as left $\mathbf {H}$ -module. Then, there exists a finite subset $\Sigma \subset \mathfrak {P}\times \mathfrak {h}^{1}$ such that $b_i 1_{c, \lambda }\in \mathscr {C}(B)1_{c, \lambda }$ is decomposed as $b_i 1_{c, \lambda } = \sum _{(c', \lambda ')\in \Sigma } 1_{c',\lambda '}b_i 1_{c,\lambda }$ for each $i\in I$ . Then, the family $\left \{ 1_{c',\lambda '}b_i 1_{c,\lambda } \right \}_{i\in I,\; (c',\lambda ')\in \Sigma }$ yields the following surjective morphism of $\mathscr {H}$ -modules:

$$\begin{align*}\bigoplus_{i\in I}\bigoplus_{(c',\lambda')\in \Sigma} \mathscr{H} 1_{c',\lambda'} \to \mathscr{C}(B) 1_{c, \lambda}.\quad \end{align*}$$

Since the left-hand side is compact, so is the right-hand side. As the family $\left \{ \mathscr {H} 1_{c, \lambda } \right \}_{(c,\lambda )\in \mathfrak {P}\times \mathfrak {h}^{1}}$ generates $\mathrm {D}^{\mathrm {b}}_{\mathrm {perf}}(\mathscr {H})$ as thick subcategory, we see that the functor preserves $\mathrm {D}^{\mathrm {b}}_{\mathrm {perf}}(\mathscr {H})$ . This proves (i).

Proof. We prove (ii). The morphism is constructed as follows: for $N\in \mathbf {H}\text{-}{\mathrm{Mod}}$ , we have

$$\begin{align*}{\mathrm{Hom}}_{\mathbf{H}}(B, N)\to {\mathrm{Hom}}_{\mathscr{H}}(\mathscr{C} (B), \mathscr{C}(N)) \end{align*}$$

by the functoriality of $\mathscr {C}$ ; it induces

$$\begin{align*}\mathscr{C}{\mathrm{Hom}}_{\mathbf{H}}(B, N) \to \bigoplus_{c, \lambda}1_{c, \lambda}{\mathrm{Hom}}_{\mathscr{H}}(\mathscr{C} (B), \mathscr{C}(N)) = \hom_{\mathscr{H}}(\mathscr{C} (B), \mathscr{C}(N)); \end{align*}$$

the natural morphism

(2.14) $$ \begin{align} \mathscr{C}{\mathrm{RHom}}_{\mathbf{H}}(B, N)\to {\mathrm{Rhom}}_{\mathscr{H}}(\mathscr{C} (B), \mathscr{C}(N)) \end{align} $$

is obtained by passing to the derived functors.

Since ${\mathrm {RHom}}_{\mathbf {H}}(B, \mathbf {H})\in \mathrm {D}^{\mathrm {b}}_{\mathrm {HC}}(\mathbf {H})$ by Proposition 2.5, the spectral completion $\mathscr {C}({\mathrm {RHom}}_{\mathbf {H}}(B, \mathbf {H}))$ is a complex of $\mathscr {H}$ -bimodules by Proposition 2.10. Consider first the case $N\in \mathrm {O}(\mathbf {H})$ . We have for $(c, \lambda )\in \mathfrak {P}\times \mathfrak {h}^{1}$

$$ \begin{align*} &1_{c,\lambda}\mathscr{C}({\mathrm{RHom}}_{\mathbf{H}}(B, N)) \cong \mathrm{L}\mathop{\varinjlim}_{l}{\mathrm{RHom}}_{\mathbf{S}}\left(\mathbf{S} / \mathfrak{m}_{c,\lambda}^l, {\mathrm{RHom}}_{\mathbf{H}}(B, N)\right) &\text{(Lemma}~{2.15})\\ & \cong \mathrm{L}\mathop{\varinjlim}_{l}{\mathrm{RHom}}_{\mathbf{H}}\left(B\otimes^{\mathrm{L}}_{\mathbf{S}}(\mathbf{S} / \mathfrak{m}_{c,\lambda}^l), N\right)& \text{(adjunction)}\\ & \cong \mathrm{L}\mathop{\varinjlim}_{l}{\mathrm{RHom}}_{\mathscr{H}}\left(\mathscr{C}(B\otimes^{\mathrm{L}}_{\mathbf{S}}(\mathbf{S} / \mathfrak{m}_{c,\lambda}^l)), \mathscr{C}(N)\right) & \text{(Theorem~1.16)} \\ & \cong \mathrm{L}\mathop{\varinjlim}_{l}{\mathrm{RHom}}_{\mathbf{S}^{\wedge}_{c, \lambda}}\left(\mathbf{S}^{\wedge}_{c, \lambda} / \mathfrak{m}_{c,\lambda}^l, {\mathrm{RHom}}_{\mathscr{H}}(\mathscr{C}(B)1_{c, \lambda},\mathscr{C}(N))\right) & \text{(adjunction)} \\ & \cong {\mathrm{RHom}}_{\mathscr{H}}(\mathscr{C}(B)1_{c, \lambda}, \mathscr{C}(N)) & \text{(Lemma~2.15)}. \end{align*} $$

Thus, (2.14) is a quasi-isomorphism for $N\in \mathrm {O}(\mathbf {H})$ . Consider now the case $N = \mathbf {H}$ . For $(c, \lambda ),(c',\lambda ')\in \mathfrak {P}\times \mathfrak {h}^{1}$ , we have

$$ \begin{align*} &1_{c,\lambda}\mathscr{C}({\mathrm{RHom}}_{\mathbf{H}}(B, \mathbf{H}))1_{c',\lambda'}\\ &\cong \mathrm{R}\mathop{\varprojlim}_{l}1_{c,\lambda}\mathscr{C}({\mathrm{RHom}}_{\mathbf{H}}(B, \mathbf{H}))1_{c',\lambda'}\otimes^{\mathrm{L}}_{\mathbf{S}^{\wedge}_{c',\lambda'}} (\mathbf{S}/\mathfrak{m}^l_{c',\lambda'}) & \text{(Lemma}~{2.15})\\ &\cong \mathrm{R}\mathop{\varprojlim}_{l}1_{c,\lambda}\mathscr{C}({\mathrm{RHom}}_{\mathbf{H}}(B, \mathbf{H}\otimes^{\mathrm{L}}_{\mathbf{S}} (\mathbf{S}/\mathfrak{m}^l_{c',\lambda'})))\\ &\cong \mathrm{R}\mathop{\varprojlim}_{l}{\mathrm{RHom}}_{\mathscr{H}}(\mathscr{C}(B)1_{c, \lambda}, \mathscr{H} 1_{c',\lambda'}\otimes_{\mathbf{S}^{\wedge}_{c',\lambda'}} (\mathbf{S}/\mathfrak{m}^l_{c',\lambda'}))& \text{(case }N\in \mathrm{O}(\mathbf{H}))\\ &\cong {\mathrm{RHom}}_{\mathscr{H}}(\mathscr{C}(B)1_{c, \lambda}, \mathrm{R}\mathop{\varprojlim}_{l}\mathscr{H} 1_{c',\lambda'}\otimes_{\mathbf{S}^{\wedge}_{c',\lambda'}} (\mathbf{S}/\mathfrak{m}^l_{c',\lambda'})))\\ &\cong {\mathrm{RHom}}_{\mathscr{H}}(\mathscr{C}(B)1_{c, \lambda}, \mathscr{H} 1_{c',\lambda'} )) = {\mathrm{RHom}}_{\mathscr{H}}(\mathscr{C}(B)1_{c, \lambda}, \mathscr{H}))1_{c',\lambda'}.& \text{(Mittag-Leffler)} \end{align*} $$

Thus, (2.14) is a quasi-isomorphism for $N = \mathbf {H}$ . Since $\mathbf {H}$ generates $\mathrm {D}^{\mathrm {b}}_{\mathrm {perf}}(\mathbf {H})$ as thick subcategory, this proves (ii).

We prove (i). Given any $\mathscr {N}\in \mathscr {H}{\mathrm {-mod}}_{\mathrm {fl}}$ , we have $\mathscr {N}\cong \mathscr {C}(N)$ for some $N\in \mathrm {O}(\mathbf {H})$ by Theorem 1.16. Hence, (ii) yields ${\mathrm {Rhom}}_{\mathscr {H}}(\mathscr {C}(B),\mathscr {N}) \cong \mathscr {C}({\mathrm {RHom}}_{\mathbf {H}}(B,N))$ . By Proposition 2.7, ${\mathrm {RHom}}_{\mathbf {H}}(B,N)$ lies in $\mathrm {D}^{\mathrm {b}}_{\mathrm {O}}(\mathbf {H})$ , so $\mathscr {C}({\mathrm {RHom}}_{\mathbf {H}}(B,N))$ lies in $\mathrm {D}^{\mathrm {b}}_{\mathrm {fl}}(\mathscr {H})$ by Theorem 1.16 again. Therefore, the functor preserves $\mathrm {D}^{\mathrm {b}}_{\mathrm {fl}}(\mathscr {H})$ .

Let $(c, \lambda )\in \mathfrak {P}\times \mathfrak {h}^{1}$ . By (ii), we have

$$\begin{align*}{\mathrm{Rhom}}_{\mathscr{H}}(\mathscr{C}(B), \mathscr{H} 1_{c, \lambda}) \cong \mathscr{C}({\mathrm{RHom}}_{\mathbf{H}}(B, \mathbf{H})) 1_{c, \lambda}\cong \mathscr{C}({\mathrm{RHom}}_{\mathbf{H}}(B, \mathbf{H}))\otimes^{\mathrm{L}}_{\mathscr{H}}\mathscr{H} 1_{c, \lambda}. \end{align*}$$

Since ${\mathrm {RHom}}_{\mathbf {H}}(B, \mathbf {H})\in \mathrm {D}^{\mathrm {b}}_{\mathrm {HC}}(\mathbf {H})$ by Proposition 2.5, we have $\mathscr {C}({\mathrm {RHom}}_{\mathbf {H}}(B, \mathbf {H}))\otimes ^{\mathrm {L}}_{\mathscr {H}}\mathscr {H} 1_{c, \lambda } \in \mathrm {D}^{\mathrm {b}}_{\mathrm {perf}}(\mathscr {H})$ by Proposition 2.11. Since the family $\left \{ \mathscr {H} 1_{c, \lambda } \right \}_{(c,\lambda )\in \mathfrak {P}\times \mathfrak {h}^{1}}$ generates $\mathrm {D}^{\mathrm {b}}_{\mathrm {perf}}(\mathscr {H})$ as thick subcategory, it follows that functor preserves $\mathrm {D}^{\mathrm {b}}_{\mathrm {perf}}(\mathscr {H})$ . This proves (i).

Proof. The first morphism is a quasi-isomorphism: as it is so for the regular A-module, also is it for every perfect complex $K\in \mathrm {D}_{\mathrm {perf}}(A)$ by dévissage. As for the second morphism, Nakayama’s lemma implies that $\varinjlim _k {\mathrm {Hom}}_A(A / \mathfrak {m}^k, L)\to L$ is an isomorphism for $L\in A{\mathrm {-mod}}_{\mathrm {fl}}$ . Given $L\in A{\mathrm {-mod}}_{\mathrm {fl}}$ , let $L\hookrightarrow I$ be an injection with I injective. We may find such I whose finitely generated submodules are of finite length. Then, by the compactness of $A / \mathfrak {m}^k$ , an easy argument of dévissage shows that $\varinjlim _k {\mathrm {Ext}}^n_A(A / \mathfrak {m}^k, L) = 0$ for $n \ge 1$ .

Proof. The first assertion results immediately from the fact that for $C, C'\in {\mathrm {Ch}}(\mathfrak {a})$ , the distance $d(C, C')$ is the number of walls separating C and $C'$ . The second can be easily derived from the first one.

Proof. We will only prove the right freeness, the proof for left freeness being similar. We prove by induction on $n\in \mathbb {N}$ the following statements for all chambers $C, C'\in {\mathrm {Ch}}(\mathfrak {a})$ :

1. (i) if $G = (C, C_1, \ldots , C_{n-1}, C')$ is a gallery, then

   $$\begin{align*}\tau_G - \tau^o_G\mathfrak{d}(G) \in F^{\mathrm{lg}}_{n - 1}{\mathrm{Hom}}_{\mathbf{A}^o}(C, C'),\quad \tau_G\mathfrak{e}(G) - 1_{C, C'}\mathfrak{d}(G) \in F^{\mathrm{lg}}_{n - 1}{\mathrm{Hom}}_{\mathbf{A}}(C, C'); \end{align*}$$

2. (ii) if $G = (C, C_1, \ldots , C_{n-1}, C')$ is a non-minimal gallery, then $\tau _G\in F^{\mathrm {lg}}_{ n-1} {\mathrm {Hom}}_{\mathbf {A}}(C, C')$ ;

3. (iii) if $G = (C, C_1, \ldots , C_{n-1}, C')$ and $G' = (C, C^{\prime }_1, \ldots , C^{\prime }_{n-1}, C')$ are minimal galleries, then $\tau _G - \tau _{G'}\in F^{\mathrm {lg}}_{ n-1} {\mathrm {Hom}}_{\mathbf {A}}(C, C')$ ;

4. (iv) the set $\{ \overline \tau _{C, C', w} \}_{\substack {w\in \hat {W} \\ d(C, w^{-1}C') = n}}$ forms a basis for

   $$\begin{align*}{\mathrm{gr}}^{\mathrm{lg}}_{n}{\mathrm{Hom}}_{\mathbf{A}}(C, C') = F^{\mathrm{lg}}_{ n} {\mathrm{Hom}}_{\mathbf{A}}(C, C') / F^{\mathrm{lg}}_{ n-1} {\mathrm{Hom}}_{\mathbf{A}}(C, C') \end{align*}$$
   as free right $\mathbf {S}$ -module;

5. (v) the quotient

   $$\begin{align*}F^{\mathrm{lg}}_{ n} {\mathrm{Hom}}_{\mathbf{A}^o}(C, C') / F^{\mathrm{lg}}_{ n} {\mathrm{Hom}}_{\mathbf{A}}(C, C') \end{align*}$$
   is right $\alpha $ -torsion-free for every $\alpha \in \Phi $ .

For $n = 0$ , all these five statements are trivial. Let $n \ge 1$ and suppose that the statements are proven for $0, \ldots , n-1$ . We prove them for n:

1. (i) Let $G^- = (C_1, \ldots , C_{n-1}, C')$ . We have

   $$\begin{align*}\tau_G - \tau^o_G\mathfrak{d}(G) = (\tau_{G^-} - \tau^o_{G^-}\mathfrak{d}(G^-))\tau_{C, C_1} + \tau^o_{G^-}(\mathfrak{d}(G^-)\tau_{C, C_1} - \tau^o_{C, C_1}\mathfrak{d}(G)). \end{align*}$$

   The term $\tau _{G^-} - \tau ^o_{G^-}\mathfrak {d}(G^-)$ lies in $F^{\mathrm {lg}}_{ n-2}{\mathrm {Hom}}_{\mathbf {A}^o}(C_1, C')$ by induction hypothesis. If $C\|_{\alpha } C_1$ for $\alpha \in \Phi $ , then $\tau _{C, C_1} = \tau ^o_{C, C_1} = \vartheta _{\alpha }$ is the Demazure operator, and thus,

   $$\begin{align*}\mathfrak{d}(G^-)\tau_{C, C_1} = \tau^o_{C, C_1}\mathfrak{d}(G^-) + \vartheta_{\alpha}\left(\mathfrak{d}(G^-) \right)s_{\alpha}. \end{align*}$$

   Since $\mathfrak {d}(G) = \mathfrak {d}(G^-)$ and $\vartheta _{\alpha }\left (\mathfrak {d}(G^-) \right )s_{\alpha }\in F^{\mathrm {lg}}_{ 0}{\mathrm {Hom}}_{\mathbf {A}^o}(C, C')$ , the second term $\tau ^o_{G^-}(\mathfrak {d}(G^-)\tau _{C, C_1} - \tau ^o_{C, C_1}\mathfrak {d}(G))$ lies in $F^{\mathrm {lg}}_{ n-1}{\mathrm {Hom}}_{\mathbf {A}^o}(C, C')$ . If $C\lceil C_1$ or $C_1\lceil C$ , then $\mathfrak {d}(G^-)\tau _{C, C_1} = \mathfrak {d}(G^-)\mathfrak {d}(C, C_1) = \mathfrak {d}(G) = \tau ^o_{C, C_1}\mathfrak {d}(G)$ , so $\tau ^o_{G^-}(\mathfrak {d}(G^-)\tau _{C, C_1} - \tau ^o_{C, C_1}\mathfrak {d}(G)) = 0$ . It follows that $\tau _G - \tau ^o_G\mathfrak {d}(G)\in F^{\mathrm {lg}}_{n-1}{\mathrm {Hom}}_{\mathbf {A}^o}(C, C')$ in all cases. The other statement can be proven similarly.

2. (ii) By (i), we have

   $$\begin{align*}\tau_G - \tau^o_G\mathfrak{d}(G) \in F^{\mathrm{lg}}_{ n - 1}{\mathrm{Hom}}_{\mathbf{A}^o}(C, C'). \end{align*}$$

   When G is non-minimal, $\tau ^o_G\in F^{\mathrm {lg}}_{ n - 1}{\mathrm {Hom}}_{\mathbf {A}^o}(C, C')$ holds by Lemma 3.2 (i); hence $\tau _G\in F^{\mathrm {lg}}_{ n - 1}{\mathrm {Hom}}_{\mathbf {A}^o}(C, C')$ . However,

   $$\begin{align*}\tau_G\mathfrak{e}(G)\in F^{\mathrm{lg}}_{ n - 1}{\mathrm{Hom}}_{\mathbf{A}}(C, C'). \end{align*}$$

   Let $\underline {\tau }_G$ be the image of $\tau _G$ in $F^{\mathrm {lg}}_{ n-1}{\mathrm {Hom}}_{\mathbf {A}^o}(C, C') / F^{\mathrm {lg}}_{ n-1}{\mathrm {Hom}}_{\mathbf {A}}(C, C')$ . It follows that $\underline {\tau }_G \mathfrak {e}(G) = 0$ . By the induction hypothesis, the space

   $$\begin{align*}F^{\mathrm{lg}}_{ n-1}{\mathrm{Hom}}_{\mathbf{A}^o}(C, C') / F^{\mathrm{lg}}_{ n-1}{\mathrm{Hom}}_{\mathbf{A}}(C, C') \end{align*}$$
   is $\alpha $ -torsion-free for each $\alpha \in \Phi $ and therefore $\mathfrak {e}(G)$ -torsion-free; hence, $\underline {\tau }_G = 0$ , and subsequently, $\tau _G\in F^{\mathrm {lg}}_{ n-1}{\mathrm {Hom}}_{\mathbf {A}}(C, C')$ holds.

3. (iii) By (i), we have $\tau _G - \tau ^o_G\mathfrak {d}(G)\in F^{\mathrm {lg}}_{ n-1}{\mathrm {Hom}}_{\mathbf {A}^o}(C, C')$ and $\tau _{G'} - \tau ^o_{G'}\mathfrak {d}(G')\in F^{\mathrm {lg}}_{ n-1}{\mathrm {Hom}}_{\mathbf {A}^o}(C, C')$ . Since G and $G'$ are minimal, we have $\mathfrak {d}(G) = \mathfrak {d}(G') = \mathfrak {d}(C, C')$ . However, the minimality also implies $\tau ^o_G = \tau ^o_{G'}$ by Lemma 3.2 (ii). Therefore,

   $$\begin{align*}\tau_G - \tau_{G'} = (\tau_G - \mathfrak{d}(G)\tau^o_G) - (\tau_{G'} - \mathfrak{d}(G')\tau^o_{G'})\in F^{\mathrm{lg}}_{ n-1}{\mathrm{Hom}}_{\mathbf{A}^o}(C, C') \end{align*}$$
   holds. However, we have $\mathfrak {e}(G) = \mathfrak {e}(G') = \mathfrak {e}(C, C')$ ; therefore, (i) yields
   $$\begin{align*}(\tau_G - \tau_{G'})\mathfrak{e}(C, C') = \tau_G\mathfrak{e}(G) - \tau_{G'}\mathfrak{e}(G')\in F^{\mathrm{lg}}_{n-1}{\mathrm{Hom}}_{\mathbf{A}}(C, C'); \end{align*}$$
   hence, the image of $\tau _G - \tau _{G'}$ in $F^{\mathrm {lg}}_{n-1}{\mathrm {Hom}}_{\mathbf {A}^o}(C, C') / F^{\mathrm {lg}}_{n-1}{\mathrm {Hom}}_{\mathbf {A}}(C, C')$ is annihilated by right multiplication by $\mathfrak {e}(C, C')$ . The induction hypothesis implies that $F^{\mathrm {lg}}_{n-1}{\mathrm {Hom}}_{\mathbf {A}^o}(C, C') / F^{\mathrm {lg}}_{n-1}{\mathrm {Hom}}_{\mathbf {A}}(C, C')$ is right $\mathfrak {e}(C, C')$ -torsion-free, so $\tau _G - \tau _{G'}\in F^{\mathrm {lg}}_{n-1}{\mathrm {Hom}}_{\mathbf {A}}(C, C')$ holds.

4. (iv) By (ii) and (iii), the space ${\mathrm {gr}}^{\mathrm {lg}}_{n}{\mathrm {Hom}}_{\mathbf {A}}(C, C')$ is spanned by $\{ \overline \tau _{C, C', w} \}_{\substack {w\in \hat {W} \\ d(C, w^{-1}C') = n}}$ as right $\mathbf {S}$ -module. This family is free – indeed, suppose there is an $\mathbf {S}$ -linear relation

   $$\begin{align*}\sum_{w}\overline\tau_{C, C', w} f_w = 0,\quad f_w\in \mathbf{S}; \end{align*}$$
   then, we have $\sum _{w}\tau _{C, C', w}f_w\in F^{\mathrm {lg}}_{ n-1}{\mathrm {Hom}}_{\mathbf {A}}(C, C')$ ; by (i), we have
   $$\begin{align*}\sum_{w}\tau^{o}_{C, C', w}\mathfrak{d}(C, w^{-1}C')f_w\in F^{\mathrm{lg}}_{ n-1}{\mathrm{Hom}}_{\mathbf{A}^o}(C, C'); \end{align*}$$
   since the family $\{ \overline \tau ^{o}_{C, C', w} \}_{\substack {w\in \hat {W} \\ d(C, w^{-1}C') = n}}$ is free over $\mathbf {S}$ by Lemma 3.2 (iii), we see that $\mathfrak {d}(C, w^{-1}C')f_w = 0$ for each w; hence, $f_w = 0$ .

5. (v) Consider the following diagram:

   

   The maps $\phi $ and $\phi '$ are the natural inclusions. By (i) and (iv), ${\mathrm {coker}}(\phi ")$ is isomorphic to

   $$ \begin{align*} \bigoplus_{w} \overline\tau^{o}_{C, C', w}\mathbf{S} / \phi(\overline\tau_{C, C', w})\mathbf{S} \cong \bigoplus_{w} \mathbf{S} / \mathbf{S}\mathfrak{d}(C, w^{-1}C'), \end{align*} $$
   and $\phi "$ is injective. Since $\mathfrak {d}(C, w^{-1}C')$ is not divisible by any element of $\Phi $ , the cokernel ${\mathrm {coker}}(\phi ")$ is right $\alpha $ -torsion-free for $\alpha \in \Phi $ . The induction hypothesis implies that ${\mathrm {coker}}(\phi ')$ is right $\alpha $ -torsion-free; hence, ${\mathrm {coker}}(\phi )$ is also right $\alpha $ -torsion-free by the snake lemma.