2.1 Abelian groups

For compact abelian Lie groups, the functor R that takes a compact abelian group to its complex representation ring can be factored into a composition of two functors as follows:

The first functor, which takes a compact abelian Lie group S to its character group ${\mathrm{Hom}(S,S^1)}$ , is a contravariant equivalence by Pontryagin duality. The second functor takes an abelian group A to the associated group ring $\mathbb Z[A]$ . In other words, $\mathrm{R}S\cong \mathbb Z[{\mathrm{Hom}(S,S^1)}]$ for any compact abelian Lie group S. For example, when T is a torus of rank r, i.e. when $T\cong (S^1)^r$ , we have $\mathrm{Hom}(T,S^1) \cong \mathbb Z^r$ . Thus, $\mathrm{R}T$ is isomorphic to $\mathbb Z[x_1^{\pm 1},\dots,x_r^{\pm 1}]$ , a Laurent ring on r generators. We will need the following consequences of this explicit description.

Under this correspondence, $\operatorname{rank}{K_S} = r - \operatorname{rank}{S}$ , the subtori of T correspond to the direct summands of $\mathbb Z^r$ , and $K_{S_1\cap S_2} = K_{S_1} + K_{S_2}$ .

2.2 Arbitrary compact groups

Textbook sources such as [Reference Bröcker and tom DieckBtD95, Reference AdamsAda69] restrict this definition to connected groups, but there is no harm in extending it to disconnected groups. Since any torus in G is necessarily contained in the identity component of G, we are effectively defining the rank of G as the rank of this connected component. In our main result, G and the subgroups $H_i$ are assumed connected, but the intersections of $H_1$ with (conjugates of) $H_2$ may not be.

1. (a) $\mathrm{R}G$ is a free as an abelian group;

2. (b) $\mathrm{R}G$ is finitely generated as a $\mathbb Z$ -algebra, hence, in particular, a noetherian ring;

3. (c) $\dim \mathrm{R}G = \operatorname{rank} G + 1$ .

For part (c), consider first a compact abelian group S. Pullback along the projection to the identity component $S\twoheadrightarrow S_0$ defines a finite ring extension $\mathrm{R}S_0\hookrightarrow \mathrm{R}S$ . Thus, $\dim{\mathrm{R}S} = \dim{\mathrm{R}S_0}$ . As $\mathrm{R}S_0 = \mathbb Z[x_1^{\pm 1}, \dots,x_r^{\pm 1}]$ with $r = \operatorname{rank} S_0 = \operatorname{rank} S$ , the claim follows in this case.

For any compact connected G, the restriction to a maximal torus T defines an integral extension $\mathrm{R}G\to\mathrm{R}T$ (see Lemma 2.6 below). Thus, the result for G follows from the result for T which we have just established. For general compact G, we similarly have an integral extension $\mathrm{R}G \to \prod_S \mathrm{R}S$ , where the product is over a finite number of compact abelian groups S (see [Reference SegalSeg68, proof of Proposition (3.2)]). Thus, $\dim\mathrm{R}G = \max_S \dim\mathrm{R}S$ . More precisely, the groups S range over all conjugacy classes of what Segal calls ‘Cartan subgroups’ of G (though note that this terminology conflicts with modern usage). Not every such Cartan subgroup is of maximal rank, as the example in the final remark of [Reference SegalSeg68, § 1] shows. However, any maximal torus of the identity component of G is a Cartan subgroup, so at least one of the groups S has maximal rank, and the claim follows.

Proof. This is [Reference SegalSeg68, Proposition 3.2].

2.3 Connected groups

We now restrict our attention to compact connected G. Recall that all maximal tori of G are conjugate, and that any element of G is contained in a maximal torus [Reference AdamsAda69, Corollaries 4.22 and 4.23]. Given a maximal torus $T\subseteq G$ , the Weyl group $\mathrm W = \mathrm W(G,T)$ is defined as $\mathrm W := (\mathrm{N}_{G}{T})/T$ . This is always a finite group. It acts on $\mathrm{R}T$ by conjugation, and we denote by $(\mathrm{R}T)^{\mathrm W}$ the subring of fixed points under this action.

1. (a) $\mathrm{R}G$ is an integral domain;

2. (b) the set of prime ideals of $\mathrm{R}T$ lying over a fixed prime ideal $\mathfrak p \subset \mathrm{R}G$ is equal to a single $\mathrm W$ -orbit of a prime in $\mathrm{R}T$ ;

3. (c) a prime ideal $\mathfrak p_T\subset \mathrm{R}T$ lying over a given prime ideal $\mathfrak p \subset \mathrm{R}G$ is unique with this property if and only if it is $\mathrm W$ -invariant; in this case, it contains every other prime ideal $\mathfrak q \subset \mathrm{R}T$ with $(i^*)^{-1}\mathfrak q \subseteq \mathfrak p$ , and $(\mathrm{R}T)_{\mathfrak p}$ is a local ring with maximal ideal $\mathfrak p_T\cdot (\mathrm{R}T)_{\mathfrak p}$ .

Claim (a) is immediate from the fact that $\mathrm{R}T$ is an integral domain and the injectivity of $i^*$ .

For claim (b), see for example [Reference Görtz and WedhornGW10, Proposition 12.27(2) and (3)]. The claim spells out the general fact that, for any commutative unital ring A, and any action of a finite group $\mathrm W$ through ring homomorphisms on A, the prime spectrum $\mathrm{Spec}(A^{\mathrm W})$ of the fixed ring can be naturally identified with $\mathrm{Spec}(A)/\mathrm W$ , the set-theoretic quotient of the prime spectrum of A. (The same statement is true in the category of affine schemes, in fact more obviously so: see [Reference Görtz and WedhornGW10, Proposition 12.27(1)]. But it is the set-theoretic statement that we need later.)

The first part of claim (c) follows immediately from claim (b). The second part follows from the first and the going-up property, which is shared by all finite (or, more generally, integral) extensions. Finally, note that the maximal primes of $(\mathrm{R}T)_{\mathfrak p}$ correspond to the primes of $\mathrm{R}T$ that lie over $\mathfrak p$ .

Similarly, for an arbitrary prime $p\in\mathbb Z$ , we can consider the ideal $\mathfrak I_pG := \operatorname{rank}^{-1}((p))$ . Again, $\mathfrak I_pT$ is a $\mathrm W$ -invariant prime ideal lying over $\mathfrak I_pG$ , so $(\mathrm{R}T)_{\mathfrak I_pG}$ is a local ring with maximal ideal $\mathfrak I_pT\cdot (\mathrm{R}T)_{\mathfrak I_pG}$ .

The inclusion of $\mathrm{R}G$ into $\mathrm{R}T$ moreover splits as follows.

Lemma 2.9. The representation ring $\mathrm{R}G$ of any compact connected G is biequidimensional, in the strong sense that all saturated chains of prime ideals have equal length, and hence satisfies the dimension formula: for any prime ideal $\mathfrak p\subset \mathrm{R}G$ ,

\[\dim \mathrm{R}G = \dim{\mathrm{R}G/\mathfrak p} + {\dim}{(\mathrm{R}G)_{\mathfrak p}}.\]

1. (i) R has a unique minimal prime ideal;

2. (ii) R is catenary, i.e. all saturated chains of prime ideals between two given prime ideals in R have equal length; and

3. (iii) R is equicodimensional, i.e. all its maximal ideals have equal height.

We now verify these two properties for $R = \mathrm{R}G$ . Property (i) clearly holds as $\mathrm{R}G$ is an integral domain (Lemma 2.6(a)). Property (ii) holds for $\mathrm{R}G$ as $\mathbb Z$ is universally catenary, as is any Dedekind domain, and as $\mathrm{R}G$ is finitely generated over $\mathbb Z$ (Lemma 2.3(b)). Property (iii) holds for $R = \mathrm{R}G$ in view of the following general fact: any integral domain R finitely generated over a Noetherian Jacobson universally catenary and equicodimensional domain S is again equicodimensional [Reference GrothendieckGro66, Proposition 10.6.1(i)]. This applies in particular to $S = \mathbb Z$ , and more generally to any Dedekind domain S with infinitely many maximal ideals.

We include here another lemma that we will use in conjunction with Lemma 2.9.

Lemma 2.10. Suppose $R\hookrightarrow S$ is a finite extension of noetherian rings, and that $\mathfrak b \subseteq S$ is some ideal. For any prime ideal $\mathfrak p\in\mathrm{Supp}_{R}{(S/\mathfrak b)}$ , there exists a prime ideal $\mathfrak p'\in\mathrm{Supp}_{R}{(S/\mathfrak b)}$ such that $\mathfrak p'\subseteq \mathfrak p$ and such that

\[ \dim{S_{\mathfrak p}} - {\dim}{(S/\mathfrak b)_{\mathfrak p}} = \dim{R_{\mathfrak p}} - {\dim}{(R/\mathfrak p')_{\mathfrak p}}. \]

2.4 Good groups and Steinberg’s theorem

For any compact connected Lie group G, the commutator subgroup G’ is semisimple [Reference SepanskiSep07, Theorem 5.22(b)]. Moreover, G can be identified with a quotient group $G\cong (G'\times T)/\Gamma$ , where T is a torus and $\Gamma$ is a finite abelian subgroup of $G'\times T$ intersecting G’ trivially [Reference SepanskiSep07, Theorem 5.22(a)]. Following Steinberg, we therefore refer to G’ as the semisimple component of G. The following is the main result of [Reference SteinbergSte75].

To avoid endlessly repeating these somewhat intricate conditions, we simply define a good Lie group to be a compact connected Lie group G that satisfies the equivalent conditions of Theorem 2.11. The result stated as Steinberg’s theorem is just the implication (a) $\Rightarrow$ (c) of this theorem combined with the following observation.

Condition (b) in Theorem 2.11 implies the following.

Condition (b) also implies that, for good G, the notions of ‘projective dimension’ and ‘weak/flat dimension’ are equivalent for finitely generated $\mathrm{R}G$ -modules. Indeed, we have the following.

We can easily sketch Pittie’s short argument for the implication (b) $\Rightarrow$ (c) based on the results and observations already included above: for a maximal torus T in a good group G, both $\mathrm{R}G$ and $\mathrm{R}T$ are regular noetherian rings (Lemma 2.13). As $\mathrm{R}T$ is in any case finitely generated as an $\mathrm{R}G$ -module (Lemma 2.4), it thus follows from [Reference SerreSer00, Chapter III.D, § 5, Theorem 13(b)] that $\mathrm{R}T$ is a projective $\mathrm{R}G$ -module. For a general subgroup of maximal rank H, we may assume that H contains T, and use holomorphic induction (Theorem 2.8) to deduce that $\mathrm{R}H$ is a direct summand of $\mathrm{R}T$ . In particular, $\mathrm{R}H$ is also projective as an $\mathrm{R}G$ -module. Finally, as we have just seen in Lemma 2.14, this is today known to imply that $\mathrm{R}H$ is free.

While Theorem 2.11 only deals with subgroups of maximal rank, the following consequence for subgroups of arbitrary ranks was proved in [Reference González-Álvaro and ZibrowiusGZ21, Corollary 3.10].

We do not reproduce the short proof of Corollary 2.16 here, but we note that besides holomorphic induction (Theorem 2.8) the main ingredient is a change-of-rings theorem for Tor. We record here two such theorems that we will have occasion to use later.

If $\mathrm{Tor}^\Lambda_*(A,\Gamma)$ vanishes in all positive degrees, then we have isomorphisms of $\Lambda$ -modules $ \mathrm{Tor}^\Lambda_i(A,C) \cong \mathrm{Tor}^\Gamma_i(A\otimes_\Lambda \Gamma, C) $ in all degrees i.

Suppose $\mathrm{Tor}_*^\Lambda(A,B)$ and $\mathrm{Tor}_*^\Sigma(B,C)$ vanish in all positive degrees. Then we have natural isomorphisms of $\Sigma\otimes\Gamma$ -modules

\[ \mathrm{Tor}_i^{\Gamma\otimes\Sigma}(A\otimes_{\Lambda}B,C) \cong \mathrm{Tor}_i^{\Lambda\otimes\Gamma}(A,B\otimes_{\Sigma}C) \]

in all degrees i.

2.5 Supports

In [Reference SegalSeg68], Segal analyses the prime spectrum $\mathrm{Spec}(\mathrm{R}G)$ of an arbitrary compact Lie group G. In particular, Segal shows that, for each prime ideal $\mathfrak p\subset \mathrm{R}G$ , there is a subgroup $S_{\mathfrak p}$ , unique up to conjugation, which is minimal with respect to the property that $\mathfrak p$ is the preimage of a prime ideal of $\mathrm{R}(S_{\mathfrak p})$ under the restriction $\mathrm{R}G\to \mathrm{R}(S_{\mathfrak p})$ . This subgroup $S_{\mathfrak p}$ , or rather any subgroup in this conjugacy class, is referred to as the support of $\mathfrak p$ . The following proposition explains how Segal’s ‘supporting subgroups’ and the usual notion of ‘prime ideals supporting a module’ are related.

1. (a) $\mathfrak p \in \mathrm{im}(\mathrm{Spec}(\mathrm{R}H) \to \mathrm{Spec}(\mathrm{R}G))$ ;

2. (b) $\mathfrak p$ contains $\ker(\mathrm{R}G \to \mathrm{R}H)$ ;

3. (c) $\mathfrak p \in \operatorname{Supp}_{\mathrm{R}G}{\mathrm{R}H}$ ;

4. (d) the support $S_{\mathfrak p}$ is conjugate to a subgroup of H.

1. (a) For subgroups $H'\subseteq H \subseteq G$ , we have $ \operatorname{Supp}_{\mathrm{R}G}{\mathrm{R}H'} \subseteq \operatorname{Supp}_{\mathrm{R}G}{\mathrm{R}H}. $

2. (b) For any two subgroups $H_1$ and $H_2$ ,

   \begin{align*} \mathrm{Supp}_{\mathrm{R}G}{(\mathrm{R}H_1 \otimes_{\mathrm{R}G} \mathrm{R}H_2)} &= \operatorname{Supp}_{\mathrm{R}G}{\mathrm{R}H_1}\cap \operatorname{Supp}_{\mathrm{R}G}{\mathrm{R}H_2}\\ &= \bigcup_{g\in G}\operatorname{Supp}_{\mathrm{R}G}\mathrm{R}(H_1\cap gH_2g^{-1}). \end{align*}

3. (c) Given a subgroup $H\subseteq G$ and a prime ideal $\mathfrak p\in\operatorname{Supp}_{\mathrm{R}G}{\mathrm{R}H}$ ,

   \[ \dim{\mathrm{R}G/\mathfrak p}\leq \operatorname{rank} H + 1. \]

4. (d) Let $\mathrm{Z} G$ denote the centre of G. If $\mathfrak p\in\operatorname{Supp}_{\mathrm{R}G}{\mathrm{R}(\mathrm{Z} G)}$ , there is exactly one prime ideal of $\mathrm{R}T$ lying over $\mathfrak p$ .

The first equality in claim (b) holds in general for finitely generated modules over a commutative ring. See [Reference Atiyah and MacdonaldAM69, Chapter 3, Exercise 19(iv)] for the general fact, and Lemma 2.4 for finite generation in the case at hand. The second equality in claim (b) again follows from the equivalence (c) $\Leftrightarrow$ (d) of Proposition 2.19.

For claim (c), recall from [Reference SegalSeg68] that the support $S_{\mathfrak p}$ is topologically cyclic, i.e. contains an element s whose powers are dense in $S_{\mathfrak p}$ . This element s is contained in some maximal torus of G, so we may as well assume $s\in T$ and hence $S_{\mathfrak p}\subseteq T$ . The ideal $\mathfrak p\subset \mathrm{R}G$ is thus the inverse image of an ideal $\mathfrak p_0\subset \mathrm{R}(S_{\mathfrak p})$ under a restriction map that factors as

\[ \mathrm{R}G \hookrightarrow \mathrm{R}T \twoheadrightarrow \mathrm{R}(S_{\mathfrak p}). \]

Recall that any inclusion of a subgroup into a torus induces a surjection on representation rings (Lemma 2.1(a)). Writing $\mathfrak p_T\subset \mathrm{R}T$ for the intermediate preimage of $\mathfrak p_0$ , we obtain induced morphisms of quotient rings as follows:

\[ \mathrm{R}G/\mathfrak p \hookrightarrow \mathrm{R}T/\mathfrak p_T \xrightarrow{\cong} \mathrm{R}(S_{\mathfrak p})/\mathfrak p_0. \]

Here, the first map is a finite extension (see Lemma 2.6), so $\dim{\mathrm{R}G/\mathfrak p} = \dim{\mathrm{R}T/\mathfrak p_T}$ , and we deduce $ \dim{\mathrm{R}G/\mathfrak p} \leq \dim{\mathrm{R}(S_{\mathfrak p})} $ . Moreover, by Lemma 2.3(c), $\dim{\mathrm{R}(S_{\mathfrak p})} = \operatorname{rank}{S_{\mathfrak p}} + 1$ , and as $S_{\mathfrak p}$ is conjugate to a subgroup of H we have $\operatorname{rank}{S_{\mathfrak p}} \leq \operatorname{rank} H$ .

For claim (d), recall that the centre of G is contained in any maximal torus. Thus, the assumption implies that $\mathfrak p\subseteq \mathrm{R}G$ can be obtained as the inverse image of some ideal in $\mathrm{R}(\mathrm{Z} G)$ under a restriction map that factors as

\[ \mathrm{R}G \hookrightarrow \mathrm{R}T \twoheadrightarrow \mathrm{R}(\mathrm{Z} G). \]

Let us again write $\mathfrak p_T$ for the intermediate preimage in $\mathrm{R}T$ , which by construction lies over $\mathfrak p$ . As the Weyl group $\mathrm W$ acts trivially on $\mathrm{Z} G$ , the restriction $\mathrm{R}T \twoheadrightarrow \mathrm{R}(\mathrm{Z} G)$ is equivariant with respect to the canonical $\mathrm W$ -action on $\mathrm{R}T$ and the trivial action on $\mathrm{R}(\mathrm{Z} G)$ . Thus, $\mathfrak p_T\subset \mathrm{R}T$ is $\mathrm W$ -invariant, and it follows from Lemma 2.6(b) that $\mathfrak p_T$ is the only ideal over $\mathfrak p$ .

For the biquotient conditions appearing in the following two examples, see the introduction or Definition 3.1 immediately below.

3.1 Reduction to diagonal

A biquotient manifold $H_1\backslash G/H_2$ can equivalently be viewed as a quotient

\[ (H_1\times H_2)\backslash\!\backslash G := (H_1\times H_2)\backslash (G\times G)/\Delta G, \]

where $\Delta G$ denotes the diagonal subgroup of $G\times G$ . Explicitly, the isomorphism is given by $g \mapsto (g,1)$ , with inverse .

1. (a) The triple $(G,H_1,H_2)$ satisfies the lax strict biquotient condition if and only if the triple $(G\times G,H_1\times H_2, \Delta G)$ does.

2. (b) The intersection ranks satisfy the equality

   \[ \mathrm{interrank}(H_1,H_2) = \mathrm{interrank}(H_1\times H_2,\Delta G). \]

3. (c) Considering $\mathrm{R}G$ as an $\mathrm{R}(G\times G)$ -module via restriction along $\Delta$ , we have canonical isomorphisms of abelian groups

   \[ \mathrm{Tor}_i^{\mathrm{R}(G\times G)}(\mathrm{R}(H_1\times H_2),\mathrm{R}G) \cong \mathrm{Tor}_i^{\mathrm{R}G}(\mathrm{R}H_1,\mathrm{R}H_2) \]
   in all degrees i. Likewise, for any prime ideal $\mathfrak p \subset \mathrm{R}G$ , we have isomorphisms of abelian groups
   \[ \mathrm{Tor}_i^{\mathrm{R}(G\times G)}(\mathrm{R}(H_1\times H_2),\mathrm{R}G)_{(\Delta^*)^{-1}(\mathfrak p)} \cong \mathrm{Tor}_i^{\mathrm{R}G}(\mathrm{R}H_1,\mathrm{R}H_2)_{\mathfrak p} \]
   in all degrees i.

4. (d) For a prime ideal $\mathfrak p \subset \mathrm{R}G$ ,

   \begin{alignat*}{13} &\text{if } &\mathfrak p&\in\mathrm{Supp}_{\mathrm{R}G}(\mathrm{R}(\mathrm{Z} G)),\\ &\text{then }& (\Delta^*)^{-1}(\mathfrak p) &\in \operatorname{Supp}_{\mathrm{R}(G\times G)}{\mathrm{R}(\mathrm{Z} (G\times G))}. \end{alignat*}

The first isomorphism of Tor-modules in claim (c) can be obtained from Theorem 2.18 by taking $\Gamma$ , $\Lambda$ , $\Sigma$ , A, B and C as indicated in the following diagram.

The Tor-modules assumed to vanish in positive degrees in Theorem 2.18 are $\mathrm{Tor}_*^{\mathbb Z}(\mathrm{R}H_1,\mathrm{R}H_2)$ and $\mathrm{Tor}_*^{\mathrm{R}G}(\mathrm{R}H_2,\mathrm{R}G)$ . In both cases, this vanishing is clear because at least one of the modules is free over the considered base ring. Thus, we obtain isomorphisms of $\mathrm{R}G\otimes\mathrm{R}G$ -modules

\[ \mathrm{Tor}_i^{\mathrm{R}G\otimes\mathrm{R}G}(\mathrm{R}H_1\otimes\mathrm{R}H_2,\mathrm{R}G) \cong \mathrm{Tor}_i^{\mathrm{R}G}(\mathrm{R}H_1,\mathrm{R}H_2) \]

in all degrees i. To simplify notation, let us temporarily write ${G^2}$ for $G\times G$ . Recall that we have natural isomorphisms $\mathrm{R}(H_1\times H_2)\cong \mathrm{R}H_1\otimes\mathrm{R}H_2$ and, in particular, $\mathrm{R}({G^2}) \cong \mathrm{R}G \otimes \mathrm{R}G$ . Under the latter isomorphism, the ring multiplication $\mathrm{R}G \otimes \mathrm{R}G\to \mathrm{R}G$ gets identified with the restriction $\Delta^*\colon \mathrm{R}({G^2}) \to \mathrm{R}G$ . We can therefore rewrite the above isomorphisms in the form asserted in the diagonal-reformulation. The second, localized claim follows immediately.

For claim (d), recall again that $\mathrm{Z} ({G^2}) = \mathrm{Z} G\times \mathrm{Z} G$ . The surjection $\Delta^*\colon \mathrm{R}({G^2})\twoheadrightarrow \mathrm{R}G$ restricts to a surjection $\mathrm{R}(\mathrm{Z} ({G^2}))\to \mathrm{R}(\mathrm{Z} G)$ , which we can view as a surjection of $\mathrm{R}({G^2})$ -modules. Localizing at $(\Delta^*)^{-1}(\mathfrak p)$ for some prime ideal $\mathfrak p$ of $\mathrm{R}G$ yields a surjection $\mathrm{R}(\mathrm{Z} ({G^2}))_{(\Delta^*)^{-1}(\mathfrak p)} \to (\mathrm{R}(\mathrm{Z} G))_{\mathfrak p}$ . Thus, if $\mathrm{R}(\mathrm{Z} G)$ is supported at $\mathfrak p$ , then $\mathrm{R}(\mathrm{Z} ({G^2}))$ must be supported at $(\Delta^*)^{-1}(\mathfrak p)$ .

3.2 The biquotient condition can be checked on finitely many intersections

By definition, the biquotient condition is a condition on infinitely many intersections $H_1\cap gH_2g^{-1}$ . We show here that only finitely many intersections need to be taken into account, and that, similarly, the intersection rank of $H_1$ and $H_2$ can be computed from only finitely many intersections. Both arguments proceed as follows: we can replace $H_1$ and $H_2$ by tori (Proposition 3.3), and for tori we need only consider conjugates by elements of the Weyl group of G (Proposition 3.5).

Suppose now that G is compact and connected, with a chosen maximal torus T and associated Weyl group $\mathrm W = (\mathrm{N}_{G}{T})/T$ as in § 2.3. Recall that $\mathrm W$ is finite. Given a subgroup $S\subseteq T$ , we write w(S) for the conjugate $hSh^{-1}$ given by an arbitrary representative $h \in \mathrm{N}_{G}{T}$ of $w\in \mathrm W$ .

The following proposition is immediate from Lemma 3.4.

Note that we can always arrange for the maximal tori $T_1$ and $T_2$ of two closed connected subgroups $H_1$ and $H_2$ to be contained in T by passing to conjugates of $H_1$ and $H_2$ if necessary.

3.3 Enlarging tori while keeping intersections almost intact

The ranks of two closed subgroups $H_1$ and $H_2$ of a compact Lie group G are clearly bounded by the rank of G in the sense that

\[ \operatorname{rank}{G} \geq \operatorname{rank}{H_1} + \operatorname{rank} {H_2} - \mathrm{interrank}(H_1,H_2). \]

The following proposition shows that, when $H_1$ and $H_2$ are tori, we can always enlarge $H_1$ in such a way that the above inequality becomes an equality, but without affecting the intersection rank.

Proof of Proposition 3.6. Choose a maximal torus T of G such that $T_1 \subseteq T$ . Replacing $T_2$ by a conjugate subgroup if necessary, we may assume that also $T_2\subseteq T$ . Then by Proposition 3.5,

\[ \mathrm{interrank}(T_1,T_2) = \max_{w\in\mathrm W}{\operatorname{rank}}{(T_1\cap w(T_2))}. \]

Let us abbreviate the ranks as follows: $r := \operatorname{rank} T$ , $r_1 := \operatorname{rank} T_1$ , $r_2 := \operatorname{rank} T_2$ , $r' := \mathrm{interrank}(T_1,T_2)$ . In view of Lemma 3.4, it suffices to construct a torus $T_1^+$ intermediate between $T_1$ and T and of rank $r_1^+ := r - r_2 + r'$ such that $ \operatorname{rank}{(T_1^+\cap w(T_2))} = r' $ for each of the finitely many Weyl group elements $w\in \mathrm W$ .

The proof works by translating the statement from subgroups of T to subgroups of $\mathbb Z^r$ to vector subspaces of $\mathbb Q^r$ , where it can be solved essentially by linear algebra. The following paragraphs provide some details.

First, let us abstract away the Weyl group and formulate our claim more generally as follows.

(T) Suppose $T_1$ and $T_2^{(1)},\dots,T_2^{(n)}$ are subtori of a torus T, with all $T_2^{(i)}$ of equal rank. Let $r := \operatorname{rank} T$ , $r_1 := \operatorname{rank} T_1$ , $r_2 := \operatorname{rank} T_2^{(i)}$ , $r' := \max_i\operatorname{rank}(T_1 \cap T_2^{(i)})$ and $r_1^+ := r-r_2+r'$ . There exists a torus $T_1^+$ such that $T_1\subseteq T_1^+ \subseteq T$ , $r_1^+ = \operatorname{rank} T_1^+$ and, for each i, $r' = \operatorname{rank}(T_1^+ \cap T_2^{(i)})$ .

Using Lemma 2.1, statement (T) may be translated into the following equivalent statement.

(Z) Suppose $K_1$ and $K_2^{(1)},\dots,K_2^{(n)}$ are direct summands of $\mathbb Z^r$ , with all $K_2^{(i)}$ of equal rank. Let $c_1 := \operatorname{rank} K_1$ , $c_2 := \operatorname{rank} K_2^{(i)}$ , $c' := \min_i\operatorname{rank}(K_1 + K_2^{(i)})$ and $c_1^- := c' - c_2$ . There exists a direct summand $K_1^-$ of $\mathbb Z^r$ such that $K_1^- \subseteq K_1$ with $c_1^- = \operatorname{rank} K_1^-$ and, for each i, $c' = \operatorname{rank}(K_1^- + K_2^{(i)})$ .

We claim that it suffices to prove the following analogous statement (Q) about rational vector spaces.

(Q) Suppose $W_1$ and $W_2^{(1)},\dots,W_2^{(n)}$ are vector subspaces of $\mathbb Q^r$ , with all $W_2^{(i)}$ of equal dimension. Let $c_1 := \dim W_1$ , $c_2 := \dim W_2^{(i)}$ , $c' := \min_i\dim(W_1 + W_2^{(i)})$ and $c_1^- := c' - c_2$ . There exists a vector subspace $W_1^- \subseteq W_1$ such that $c_1^- = \dim W_1^-$ and, for each i, $c' = \dim(W_1^- + W_2^{(i)})$ .

To deduce statement (Z) from statement (Q), note that there is a close relation between vector subspaces of $\mathbb Q^r$ and subgroups of $\mathbb Z^r$ . For a vector subspace $W\subseteq \mathbb Q^r$ , the intersection $W\cap \mathbb Z^r$ is a direct summand of $\mathbb Z^r$ , of rank equal to the dimension of W. Conversely, from any subgroup $K\subseteq \mathbb Z^r$ , we can obtain the vector subspace $K\otimes_{\mathbb Z}\mathbb Q \subseteq \mathbb Q^r$ of dimension equal to the rank of K. We obtain in this way a bijection between vector subspaces of $\mathbb Q^r$ and direct summands of $\mathbb Z^r$ . Moreover, for arbitrary subgroups $K_1, K_2 \subseteq \mathbb Z^r$ , we have $(K_1 + K_2)\otimes\mathbb Q = K_1\otimes\mathbb Q + K_2 \otimes \mathbb Q$ . (Note that $K_1 + K_2$ may not be a direct summand even when $K_1$ and $K_2$ are.) Thus, given direct summands $K_1$ and $K_2^{(i)}$ as in statement (Z), we may consider the vector spaces $W_1 := K_1\otimes\mathbb Q$ and $W_2^{(i)} := K_2^{(i)}\otimes\mathbb Q$ in statement (Q), obtain a vector space $W_1^-\subseteq W_1$ , and then take $K_1^- := W_1^-\cap \mathbb Z^r$ . As noted, this is a direct summand of $\mathbb Z^r$ , clearly contained in $K_1$ , and the ranks work out as intended.

Next, we reduce statement (Q) to the case $n=1$ .

(Q $^\circ$ ) Let $W_1$ and $W_2$ be vector subspaces of $\mathbb Q^r$ , with $c_i := \dim W_i$ . For any integer $c_1^-$ such that $0\leq c_1^-\leq \dim(W_1 + W_2) - c_2$ , there exists a vector subspace $W_1^-\subseteq W_1$ of dimension $c_1^-$ such that $\dim(W_1^- + W_2) = c_1^- + c_2$ . Moreover, the vector subspaces $W_1^-$ with these properties form a (non-empty) Zariski-open subset of the rational Grassmannian $\mathrm{Gr}_{\mathbb Q}({c_1^-},{W_1})$ .

To deduce statement (Q) from statement (Q $^\circ$ ), note that for each i we have $c_2 \leq c' \leq \dim(W_1 + W_2^{(i)})$ in statement (Q), so $0 \leq c_1^- \leq \dim(W_1 + W_2^{(i)}) - c_2$ . Thus, by statement (Q $^\circ$ ), for each i the vector subspaces $W_1^-\subseteq W_1$ with $\dim(W_1^- + W_2^{(i)}) = c'$ form a non-empty Zariski-open subset of $\mathrm{Gr}_{\mathbb Q}({c_1^-},{W_1})$ . Thus, we can pick any $W_1^-$ that lies in the intersection of these finitely many Zariski-open subsets.

Finally, statement (Q $^\circ$ ) can be phrased more elegantly, and proved, by noting that $\dim(W_1+W_2) - c_2 = c_1 - \dim(W_1 \cap W_2)$ , and by noting that the condition on the dimension of $W_1^-+W_2$ is equivalent to the condition $W_1^- \cap W_2 = \{0\}$ . Such a $W_1^-$ exists because we can simply choose a complement W’ of $W_1\cap W_2$ in $W_1$ , and then pick an arbitrary subspace $W_1^- \subseteq W'$ of dimension $c_1^-$ . To see that trivial intersection with $W_2$ is an open condition on $\mathrm{Gr}_{\mathbb Q}({c_1^-},{W_1})$ , note that this condition on $W_1^-$ can be rephrased as follows. The restriction of the natural projection $W_1\twoheadrightarrow W_1/(W_1\cap W_2)$ to $W_1^-$ is a monomorphism. By change of coordinates, we may assume without loss of generality that this projection is the projection onto the first $\dim(W_1+W_2) - c_2$ coordinates. Then the restriction is a monomorphism if and only if, for a given basis $(b_1,\dots,b_{c_1^-})$ of $W_1^-$ , one of the maximal minors is non-zero. These $(c_1^-\times c_1^-)$ -minors are precisely the Plücker coordinates on the Grassmannian, so this is an open condition.

3.4 Reduction to toroidal subgroups

In case $H_1$ and $H_2$ are good, Theorem 2.11 can be used to show that the implication of the red:torus is even an equivalence, but we do not need this.