On the Resolution of Reductive Monoids and Multiplicativity of $\gamma$-Factors

Every theory of $L$-functions must satisfy the axiom of multiplicativity/inductivity, which simply requires that $\gamma$-factors for induced representations are equal to those of the inducing representations. This axiom is a theorem for Artin $L$-functions and those obtained from the Langlands-Shahidi method [Sha10], and is a main tool in computing $\gamma$-factors, root numbers, and $L$-functions. On the other hand, its proof in the cases obtained from Rankin-Selberg methods are quite involved and complicated. It is also central in proving equality of these factors when they are defined by different methods and in establishing the local Langlands correspondence (LLC) [Sha12, Sha17, HT01, Hen00, GT11, CST17] . Its importance as a technical tool in proving certain cases of functoriality [CKPSS04, Kim03, KS02] is now well established.

In this paper we will provide a proof of multiplicativity for $\gamma$-factors defined by the method of Braverman-Kazhdan/Ngo [BK02, BK10, BNS16, Ngô20] and L. Lafforgue [Laf14] in general under the assumption that the $\rho$-Fourier transforms on the group $G$ and the inducing Levi subgroup $L$ commute with the $\rho$-Harish-Chandra transform, a generalized Satake transform sending $C_{c}^{\infty}(G(k))\to C_{c}^{\infty}(L(k))$, where $\rho$ is a finite dimensional representation of the $L$-group of $G$ by means of which the $\gamma$-factors are defined.

Within our proof, we define a space $\mathcal{S}^{\rho}(G)$ of $\rho$-Schwartz functions for every $\rho$ as

This definition is crucial since the $\rho$-Schwartz functions defined in this way will be uniformly bi-$K$-finite (see equation (5.16) and Lemma $5.5$), making the descent to the inducing level possible, an important step in the proof of multiplicativity. While the $\gamma$-factor can be defined as the kernel of the Fourier transform, it is the full functional equation that allows our descent to the inducing level in a transparent fashion, using our definition of $\rho$-Schwartz functions.

In [BK10], Braverman and Kazhdan defined their Schwartz space as a ”saturation” of ours. But our Schwartz space, which is denoted by $V_{\rho}$ in [BK10], covers a significant part of theirs and in particular, contains the $\rho$-basic function as we prove in Proposition $5.3$ This is done using the extended Satake transform to almost compact functions [Li17] and the fact that it commutes with the Fourier transform induced from tori which is now defined in general, cf. Section $6$ and in partiular diagram $(6.8)$.

The commutativity assumption allows us to extend the $\rho$-Harish-Chandra transform to $\mathcal{S}^{\rho}(G)$, commuting with $J^{\rho}$ and $J^{\rho_{L}}$, respectively, where $\rho_{L}$ is the restriction of $\rho$ to the $L$-group of $L$. This construction of $\mathcal{S}^{\rho}(G)$ agrees with that of Braverman-Kazhdan in the case of doubling method [BK02, GPSR87, Li18, LR05, PSR86, Sha18, JLZ20, GL20], since $G$ being the interior of the defining monoid embeds as a unique open orbit into the Braverman-Kazhdan space (cf. [Li18]). Our proof is a generalization of Godement-Jacquet for $GL_{n}$, Theorem 3.4 of [GJ72].

One expects $\mathcal{S}^{\rho}(G)\subset C_{c}(M^{\rho}(k))$, where the latter is defined as the space of functions of compact support on $M^{\rho}(k)$, the monoid attached to $\rho$ whose restriction to $G(k)$ are smooth (locally constant) and sending $C_{c}^{\infty}(G(k))$ to itself inside $C_{c}(M^{\rho}(k))$. The group $G$ being smooth as a variety, the singularities of the monoid $M^{\rho}$ are outside of $G$. Renner’s construction (Section 2) realizes $M^{\rho_{L}}\subset M^{\rho}$ as a closed subvariety for any Levi subgroup $L$ that contains $T$, where $T$ is a fixed maximal torus of $G$, used in the construction of $M^{\rho}$.

The $\rho$-Harish-Chandra transform cannot be defined on $C_{c}(M^{\rho}(k))$ since the modulus character $\delta_{P}$ may vanish outside $L(k)\subset M^{\rho_{L}}(k)$, but it can be defined on the image of $\mathcal{S}^{\rho}(G)$ inside $C_{c}(M^{\rho}(k))\cap C^{\infty}(G(k))$, that lands in the image of $\mathcal{S}^{\rho_{L}}(L)$ in $C_{c}(M^{\rho_{L}}(k))$ by the above discussion.

Our commutativity axiom, which implies multiplicativity and multiplicativity itself give rise to an inductive scheme that allows for a definition of Fourier transform $J^{\rho}$ by building from the case of conjugacy classes of Levi subgroups $L$ of $G$. In fact, Theorem 5.3 gives the $\gamma$ factors $\gamma(s,\pi,\rho,\psi)$, $\pi$ an irreducible constituent of $\mathrm{Ind}(\sigma)$, equal to the inducing $\gamma$-factor $\gamma(s,\sigma,\rho_{L},\psi)$, which in turn is defined through convolution by $J^{\rho_{L}}$. For example, for $GL_{2}$, the Levi subgroups consist of split tori for which a canonical Fourier transform exists (c.f. [Ngô20]; see (6.2) here) and $GL_{2}$ itself, which is equivalent to understanding supercuspidal $\gamma$-factors. We refer to section 5.4 for a more detailed discussion of this inductive construction.

In the case of $GL_{2}$, Laurent Lafforgue [Laf14] has defined a candidate distrubution which is shown formally to commute with the Harish-Chandra transform and evidence exists that it may give the correct supercuspial factors as observed by Jacquet, but it is still unknown if this is the right distribution. Work in this direction for tamely ramified representations is being pursued by the second author.

Although our definition of the space $\mathcal{S}^{\rho}(G)$ depends on the knowledge of how $J^{\rho}$ acts on $C_{c}^{\infty}(G(k))$, this seems to be the most efficient way of defining $\mathcal{S}^{\rho}(G)$ at present and sufficient for our purposes as a working definition, allowing us to begin making some initial steps toward understanding the general theory, and as observed earlier after equation (0.1), essential in proving the uniform $K$-finiteness of $\rho$-Schwartz functions.

One hopes that the geometry of $M^{\rho}$ will provide some insight into what this Fourier transform ought to be. In fact, the geometric techniques used to study the basic functions on reductive monoids via arc spaces in the function field setting [BNS16] tells us that the nature of the singularities of the monoid very much controls the asymptotics of the basic function. Taking cue from this, it is natural to consider the geometry of the singularities in the $p$-adic case as well. As a first step, we may classify the singularities of our monoids via the theory of spherical varieties and we find that there is a good and explicit choice of $G$-equivariant resolution of singularities [Bri89, Per14]. The resolution is moreover rational and so we may pass without trouble between differential forms on the monoid and its resolution. The geometric aspects of this theory are discussed in part in Section 3 of the present paper. Since our Schwartz spaces are, at least tentatively, linked by the definition of the Fourier transform $J^{\rho}$ via $\mathcal{S}^{\rho}(G)=C_{c}^{\infty}(G(k))+J^{\rho}(C_{c}^{\infty}(G(k))$, we are able, at least speculatively, to unite the themes of this paper. Here is the outline of the paper.

Section 1 is a quick review of the method for $GL(n)$ as developed in [GJ72]. Renner’s construction of reductive monoids is briefly discussed in Section 2 which concludes with a treatment of the cases of symmetric powers for $GL(2)$, describing all the objects involved in those cases. Section $3$ covers the geometric aspects studied in the paper. This includes the resolution of the singularities of reductive monoids, leading to a proof of rationality of these singularities. This allows a transfer of measures from the monoid to its resolution as discussed in Section $4$ and can be applied to the integration of basic functions on corresponding toric varieties in Example $4.1$. Multiplicativity is stated and proved in Section $5$, concluding with the example of $GL(n)$ in $5.3$ and a discussion of the inductive nature of Fourier transforms in $5.4$. In proving multiplicativity, we have found it easier to work with the full functional equation rather than the definition given by convolutions. The cases of a tori and unramified data are addressed in Section $6$. The paper is concluded with a brief discussion of the doubling construction of Piatetski-Shapiro and Rallis with relevant references cited.

The authors would like to thank J. Getz, D. Jiang, and B.C. Ngo for helpful conversations. A part of this paper was presented by the first author during the month long program “On the Langlands Program: Endoscopy and Beyond” at the Institute for Mathematical Sciences, National University of Singapore, December 17, 2018-January 18, 2019. He would like to thank the Institute and the organizers: W. Casselman, P.-H. Chaudouard, W.T. Gan, D. Jiang, L. Zhang, and C. Zhu, for their invitation and hospitality. Finally, we would like to thank Jayce Getz, Chun-Hsien Hsu, and Michel Brion for their comments after the paper was posted on arXiv. Both authors were partially supported by NSF grants DMS 1801273 and DMS 2135021

We recall that the Godeement–Jacquet [GJ72] theory for standard $L$-functions of $GL_{n}$, which this method aims to generalize, can be presented briefly through the definition of corresponding $\gamma$-factors.

Let $F$ be a $p$-adic field and $G=GL_{n}$. Let $\pi$ be an irreducible admissible representation of $GL_{n}(F)$. Given a Schwartz function $\phi$ on $M_{n}(F)$, i.e., $\phi\in C^{\infty}_{c}(M_{n}(F))$, a smooth function of compact support on $M_{n}(F)$, one can define a zeta-function

where $f(x)=\langle\pi(x)v,\widetilde{v}\rangle,\ v\in\mathcal{H}(\pi)$ and $\widetilde{v}\in\mathcal{H}(\widetilde{\pi})$, is a matrix coefficient and $s\in\mathbb{C}$. Here $\widetilde{\pi}$ is the contragredient of $\pi$. Let

be the Fourier transform of $\phi$ with respect to the (additive) character $\psi\neq 1$ of $F$.

If $\check{f}(g)=f(g^{-1})$, $g\in GL_{n}(F)$, then we can consider $Z(\hat{\phi},\check{f},s)$. The Godement–Jacquet theory defines a $\gamma$-factor $\gamma^{\text{std}}(\pi,s)$ which depends only on $\pi$ and $s$ and is a rational function of $q^{-s}$, satisfying

for all $\phi$ and $f$.

It is not hard to see that if we introduce the Int$(G)$-invariant kernel, $G=GL_{n}$,

of the Fourier transform, then

by virtu of irreducibility of $\pi$ and the Schur’s lemma.

This formulation for the $\gamma$-factor is a quick and convenient way of introducing them which is amenable to generalization. We can therefore write

pointing to the significance of the kernel $\Phi_{\psi}$ in defining the $\gamma$-factors.

To treat the general case we need to generalize $M_{n}(F)$. Let $k$ be an algebraically closed field of characteristic zero. A monoid $M$ is an affine algebraic variety over $k$ with an associative multiplication and an identity 1. For our purposes, we also need $M$ to be normal, i.e., $k[M]$ is integrally closed in $k(M)$. We can always find a normalization in case $M$ is not normal, i.e., an epimorphism $\widetilde{M}\to M$ such that integral closure of $k[M]$ in $k(M)$ equals $k[\widetilde{M}]$ as we realize $k[M]\hookrightarrow k[\widetilde{M}]$.

We thus let $M$ be a normal monoid and let $G=G(M)=M^{*}$, be the units of $M$. We say $M$ is reductive if $G$ is. We now like to attach a monoid to a finite dimensional representation $\rho$ of $\hat{G}={{}^{L}}G$, $L$-group of $G$, $\rho:\hat{G}\to GL(V_{\rho})$, where $G$ is a split reductive group. Let $T\subset G$ be a maximal torus and write

where $W(\rho)$ is the set of weights of $\rho$. Let $\Lambda=\text{Hom}(\mathbb{G}_{m},T)$ be the set of cocharacters of $T$ or characters of $\hat{T}$ and set $\Lambda_{\mathbb{R}}=\Lambda\otimes{{}_{\mathbb{Z}}}\mathbb{R}$. Next, denote by $\Omega(\rho)$ the convex span in $\Lambda_{\mathbb{R}}$ of weights of $\rho$ and let $\xi(\rho)$ be the cone in $\Lambda_{\mathbb{R}}$ generated by rays through $\Omega(\rho)$.

Let $\sigma^{\vee}=\xi(\rho)^{\vee}\cap X^{*}(T)$, be the “rational” dual cone to $\xi(\rho)\cap X_{*}(T)$ and $k[\sigma^{\vee}]$ the group algebra of $\sigma^{\vee}$. One can then identify $\sigma^{\vee}$ as a subset of $k[\sigma^{\vee}]$ by $\mu\in\sigma^{\vee}$ going to $\chi_{\mu}$, defind by

$\chi_{\mu}(\mu)=1$, and $\chi_{\mu_{1}}\cdot\chi_{\mu_{2}}=\chi_{\mu_{1}+\mu_{2}}$, where the sum is the one on the semigroup $\sigma^{\vee}$. We note that this is valid for any semigroup $S$ and

Now, assume $G$ has a character

such that

sends $z\in\mathbb{C}^{*}$ to $z\cdot\text{Id}$. This means that $\langle\nu,\omega\rangle=1$ for any weight $\omega$ of $\rho$. In fact, for $z\in\mathbb{C}^{*}$,

and thus $\langle\nu,\omega\rangle=1$. Then $\nu\in\sigma^{\vee}$ and its existence implies that $\xi(\rho)$ is strictly convex, i.e., has no lines in it. In fact, the cone $\xi(\rho)$ is contained in the open half–space of vectors $x\in\Lambda_{\mathbb{R}}$, $\Lambda_{\mathbb{R}}=\Lambda\otimes_{\mathbb{Z}}\mathbb{R}$, $\Lambda=\text{Hom}(\mathbb{G}_{m},T)$, satisfying $\langle\nu,x\rangle>0$. It is therefore strictly convex. (cf. [Ngô20],Proposition 5.1).

By the theory of toric varieties [CLS11], the strictly convex cone $\xi(\rho)$ determines (uniquely) a normal toric variety, i.e., a normal affine torus embedding $j:T\subset M_{T}$. Here $M_{T}$ is the monoid for $T$ attached to $\rho|\hat{T}$. More precisely, $M_{T}=\text{Spec}(k[\sigma^{\vee}])$ by Theorem 1.3.8, pg. 39 of [CLS11]. By definition 3.19 of [Ren05], $k[\sigma^{\vee}]$ is generated by $X(M_{T})$ and thus $X(M_{T})=\sigma^{\vee}$, the semigroup defining $M_{T}$. The embedding $j:T\subset M_{T}$, defines $j^{*}:X(M_{T})\hookrightarrow X(T)$, a morphism of semigroups, into the character group of $T$.

The dominant characters in $X(T)$ all lie in $X(M_{T})$ and are those that extend to semigroup morphism $M_{T}\to\mathbb{A}^{1}=\mathbb{G}_{a}$ (Proposition 3.20 of [Ren05]).

Finally we observe that $\nu$ is integral and dominant and thus $\nu\in X(M_{T})$.

To proceed, we remark that the Weyl group $W=W(G,T)$ acts on $T,M_{T},X(T)$ and $X(M_{T})$ in the usual manner. Thus the dual rational cone $\sigma^{\vee}$ may be identified with $X(M_{T})$, both semigroups, since its group algebra generated by elements of $X(M_{T})$ or $\sigma^{\vee}$, is $k[M_{T}]$ as we discussed earlier.

Let $\lambda\in X(T)$ be a dominant (and integral) character. Then $\lambda|T_{\text{der}}$ defines an irreducible finite dimensional (rational) representation $\mu^{\circ}_{\lambda}$ of $G_{\text{der}}$, $T_{\text{der}}=T\cap G_{\text{der}}$, of highest weight $\lambda|T_{\text{der}}$. Since

we can extend $\mu^{\circ}_{\lambda}$ to an irreducible rational representation $\mu_{\lambda}=\mu^{\circ}_{\lambda}\otimes(\lambda|Z(G))$ of

This in particular is valid for dominant elements in $X(M_{T})$. We note that $\nu\in X(M_{T})$ is one such.

Now choose $\{\lambda_{i}\}^{s}_{i=1}$ so that $\bigcup^{s}_{i=1}W\!\cdot\!\lambda_{i}\subset X(M_{T})$ generates $X(M_{T})$. Let $(\mu_{\lambda_{i}},V_{\lambda_{i}})$ be the representation attached to $\lambda_{i}$. Set $\mu=\bigoplus\limits^{s}_{i=1}\mu_{\lambda_{i}}$ and $V=\bigoplus\limits^{s}_{i=1}V_{\lambda_{i}}$. The character $\nu$ will be among these $\lambda_{i}$. We may assume $\lambda_{1}=\nu$. Define $M_{1}=\overline{\mu(G)}\subset\text{End}(V)$. We let $M$ be a normalization of $M_{1}$.

We note that we may take $\lambda_{i}|T_{\text{der}}$ to be among the fundamental weights of $G_{\text{der}}$, with $\mu_{\lambda_{1}}=\nu$ extending the trivial representation of $G_{\text{der}}$ since $\nu$ is a representation (character) of $G/G_{\text{der}}\simeq Z(G)/G_{\text{der}}\cap Z(G)$.

Renner’s classification of reductive monoids uses the “extension principle” [Ren05]. The extension principle follows in the spirit of many similar classification results for spherical varieties that rely on the existence of an open $B\times B^{\textrm{op}}$-orbit where $B$ is a Borel subgroup of $G$, that is in Renner’s case adapted to account for the monoid structure. By Renner’s classification, the category of Reductive monoids is equivalent to the category of tuples $(G,T,\overline{T})$, where $T$ is any maximal torus in $G$ and $\overline{T}$ is a Weyl-group stable toric variety. A morphism of data $(G,T,\overline{T})\to(G^{\prime},T^{\prime},\overline{T^{\prime}})$ are given by a pair $(\varphi,\tau)$ where $\varphi:G\to G^{\prime}$ is a morphism of reductive groups and $\tau:\overline{T}\to\overline{T^{\prime}}$ a morphism of toric varieties such that the restriction of each morphism to the maximal torus agree $\varphi|T=\tau|T$. In the following, we reframe these results in terms of the theory of spherical varieties, in order to state the existence of a $G$-equivariant resolution of singularities [BK07], [Rit03].

Let $G$ be a split reductive group defined over a characteristic zero field $k$. Let $X$ be a variety defined over $k$ with an rational action $\alpha:G\times X\to X$. In this case we say $X$ is a $G$-variety. Let $\mathscr{O}_{X}$ be the sheaf of regular on $X$. If $X$ is affine, we will identify $\mathscr{O}_{X}$ with the coordinate algebra $k[X]$. In this case $\alpha$ induces as usual a co-action map $\alpha^{*}:k[X]\to k[G]\otimes k[X]$ by $(\alpha^{*}f)(x,g)=f(g^{-1}x)=\sum_{i}h_{i}(g)f_{i}(x)$ with $h_{i}\in k[G]$, where the latter is a finite sum. Thus each $f$ determines a finite dimensional $G$-module. Because $G$ is reductive and we are in characteristic $0$, each finite dimensional $G$-module decomposes as a finite sum of irreducible representations indexed by their highest weight vector with weight $\lambda$. As such we may decompose $k[X]=\bigoplus k[X]_{\lambda}$, indexed by the $\lambda$ that appear in $k[X]$.

Suppose $X$ is spherical. Then as above, by highest weight theory, each dominant integral character $\lambda$ of $T(k)$ that appears in $k[X]$ has a highest weight vector $f_{\lambda}$. The line $k\cdot f_{\lambda}$ is the unique line stabilized by $B$ on which $B$ acts through the character $\lambda:f_{\lambda}(bx)=\lambda(b)f_{\lambda}$. In other words $f_{\lambda}$ is a semi-invariant. Suppose $f_{1}$ and $f_{2}$ are semi-invariants that are $\lambda$-eigenfunctions appearing in $k[X]$. Then the rational function $f_{1}/f_{2}$ is $B$-invariant. As the $B$-orbit in $G$ is dense, this implies $f_{1}/f_{2}$ is constant. Hence for spherical varieties, each $\lambda$ that appears can only appear with multiplicity one. For general reductive groups actions, even the “naive” (categorical) quotient is reasonably well behaved.

We are interested in reductive monoids, which have open $G$-orbit and are spherical with respect to $G\times G$ with an open dense borel $=B\times B^{\mathrm{op}}$ orbit.

Therefore such $X$ are simple. Once again let us consider an affine simple $X$ as a $G\times G$ variety. Decomposing $\bigoplus_{\lambda}k[X]_{\lambda}\cong V_{\lambda}$ where $V_{\lambda}$ is the highest weight module for $\lambda$. The $(B,\lambda)$ eigenfunction $f_{\lambda}$ is $U$-invariant. Thus one may consider taking $U\times U^{\mathrm{op}}$-invariants $k[X]^{U\times U^{\textrm{op}}}$ are therefore generated as a vector space by the $f_{\lambda}$. Using the following

We can conclude that the variety $X/(U\times U^{\mathrm{op}})$ is a $T\cong U\backslash BB^{\mathrm{op}}/U^{\mathrm{op}}\hookrightarrow U^{\mathrm{op}}\backslash G/U$ variety, on which $T$ acts on $f_{\lambda}$ through the character $\lambda$. In other words, we have a ring $\bigoplus(V_{\lambda}\otimes V_{\lambda}^{*})^{(U\times U^{\textrm{op}})}$ graded by $\Lambda_{X}=\{\lambda\in X^{*}(T):\ k[X]_{\lambda}\neq 0\}$. By Theorem $3.5$, this is a finitely generated monoid. Each summand has a diagonalizable action by the torus $T$, giving an equivalent characterization of toric varieties: hence defines an affine $T$ embedding. If $X$ is a reductive monoid, this must therefore be equal to Renner’s cone. Moreover, if $X$ is normal the associated toric variety is normal, hence the cone of weights of $X$ defining the toric variety is saturated.

Recall that a $G$-variety is simple if it contains a unique closed orbit. When $X$ is affine, it is enough that $G$ embeds as an open subvariety to imply $X$ is simple. We state without proof the following:

To classify a general spherical variety one needs the following additional data.

Briefly (although see [Kno91] for details) a simple spherical variety $X$ is determined by its weight monoid, plus the data of which $B$-stable boundary divisors containing the unique closed orbit $Z$ are $G$-stable and which are not. More precisely, each divisor $D$ defines a valuation by first restricting $D\cap G$ which defines a valuation on the multiplicative group of rational functions $k(G)^{\times}$ on $G$ (the valuation $v_{D}\cap G$ is the order of vanishing of a rational function $f/g$ on $D\cap G$). This defines a so-called colored cone, defined by the $\mathbb{Q}_{\geq 0}$ span of the finite number of valuations $v_{D}$ as above in which the colors are the valuations that are $B$-stable but not $G$-stable (or dually, their corresponding divisors have a $B$-open-orbit).

Thus the set $D(X:Z)$ gives the set of colors of the simple spherical variety $X$. The cone generated by $\mathcal{B}(X)$ and the natural image of $D(X:Z)$ in the set of $K[G]$ valuations is the colored cone $\mathcal{C}$ determined by the data $(V(X),\mathcal{B}(X))$ that determines up to isomorphism the spherical variety $X$. For reductive monoids, this cone is equivalent to the one constructed in the earlier section via highest weight theory.

Thus, for a monoid with reductive group $G$ embedded as its unit group, the colors of $M$ are all the $B\times B^{\text{op}}$ stable irreducible divisors of $G$, and thus the monoid is determined purely by the data $\mathcal{B}(M)$ or equivalently $\mathcal{C}(M)$. We state for convenience this form of the classification.

The theory of reductive monoids affords us an explicit description of $\mathcal{B}(M)$ purely in monoid-theoretic terms.

As a consequence of the above isomorphism, the $L$ orbits of $Z$ correspond to $G$ orbits in $X$. Note that when $X=M$ is a reductive monoid, this is precisely Renner’s extension theorem [Ren05] with $P_{M}=B\times B^{\text{op}}$ and $Z=\overline{T}$. The proposition implies that the singularities of $X$ are those determined by the cone of the toric variety $Z$.

Let us recall Renner’s extension theorem for normal reductive monoids. It states that a morphism of reductive monoids $M\to M^{\prime}$ is given by the data $(G,T,\overline{T})$ and $(G^{\prime},T^{\prime},\overline{T^{\prime}})$ and a morphism $\varphi:M\to M^{\prime}$ is equivalent to $\varphi|G\to G^{\prime}$ and $\tau|\overline{T}:\overline{T}\to\overline{T^{\prime}}$. Briefly, in $M$ one has an analogue of the open cell which is the image of an open embedding $U^{\mathrm{op}}\times\overline{T}\times U\to U^{\mathrm{op}}\overline{T}U$ which has codimension $\geq 2$ in $M$. Thus one gets an equivariant morphism $(u^{\prime},t,u)\mapsto\varphi(u^{\prime})\tau(t)\varphi(u)\in M^{\prime}$. By normality of $M$, the codimension $\geq 2$ condition extends the map uniquely to $M\to M^{\prime}$, and one verifies this is in fact a morphism of monoids.

In view of the above structure theory, the singularities of a spherical $G$ variety $X$ are determined by those of its associated toric variety $X//U\cong\overline{T}$. Smooth toric varieties are classified as follows.

As a general toric variety is glued from affine toric varities, it is given by a fan consisting of rational polyhedral cones. Thus a toric variety is smooth if and only if its fan consists of rational polyhedral cones whose extremal rays generate (as a $\mathbb{Z}$-module) $X^{*}(T)$. Moreover any peicewise linear morphism of rational polyhedral cones $\mathcal{C}\to\mathcal{C}^{\prime}$ defines a $T$-equivariant morphism of toric varietie, thus, by Theorem $3.18$, one obtains an algorithm giving a resolution of singularities of a toric variety $\overline{T}$.

Finally, one may define a $G$-equivariant resolution from Remark $3.17$ and Theorems $3.18$ and $3.19$:

Next we discuss the rationality of singularities. Recall that $X$ has rational singularities if a (and hence any) resolution $r:\widetilde{X}\to X$ has vanishing higher cohomology: $R^{i}r_{*}\mathcal{O}_{\widetilde{X}}=0$ if $i>0$. Moreover, recall that for a resolution $r:\widetilde{X}\to X$ the fiber over the singular locus $X^{\mathrm{sing}}=r^{-1}(X^{\mathrm{sing}})=E$ is the exceptional divisor.