Let $G$ be a reductive algebraic group with choices of Borel subgroup and maximal torus $B \supseteq T$ and unipotent radical $U$ over an algebraically closed field $k$ of characteristic zero. Is the affine closure $\overline{G/U} = \operatorname{Spec}(A)$ Cohen–Macaulay? If so, is there a reference to this fact? If not, is there a reference to a specific counterexample?


Note: We have that $A$ is Cohen—Macaulay when $G = \operatorname{SL}\_2$ since $\overline{G/U} \cong \mathbb{A}^2$ is smooth. However, in the paper [Popov - Contractions of Actions of Algebraic Groups](https://www.researchgate.net/publication/252438955_Contractions_of_actions_of_reductive_algebraic_groups), the author seems to suggest that $A$ is not Cohen–Macaulay in general. Specifically, in section 6, Popov introduces the notion of a stable property of local rings of points of schemes, and explicitly states that "Among the examples of properties (P) of open type given in section 5 [which includes the property of being Cohen–Macaulay], the following properties are stable:" and goes on to give a list of properties which does not include the property of being Cohen–Macaulay. Furthermore, following Popov, an open property is stable if and only if it is closed under the invariants of a reductive group and the property is stable under tensor product of regular functions on the basic affine space of reductive groups, and the invariants of a reductive group acting on a Cohen–Macaulay ring is Cohen–Macaulay by the Hochster–Roberts theorem. If $A$ is not Cohen–Macaulay in general, I am interested in a specific counterexample.


Edit: For $\operatorname{SL}\_3$, we have $A \cong k[a,b,c,x,y,z]/(ax + by + cz)$ which is Cohen–Macaulay. To see this, note that using [this problem set](https://www.mit.edu/%7Efengt/Braverman_1.pdf) ([Wayback Machine](https://web.archive.org/web/20200328172550/https://www.mit.edu/%7Efengt/Braverman_1.pdf)), the algebra of functions $A$ is the quotient of the free polynomial algebra in six variables (three coming from the standard representation $V(\omega\_1)$ and three coming from the dual $V(\omega\_2)$, respectively) modulo the quadratic relation given by the multiplication rule $V(\omega\_1) \otimes V(\omega\_2) \cong V(\omega\_1 + \omega\_2) \oplus V(0) \to V(\omega\_1 + \omega\_2)$, where the rightmost arrow is the quotient arrow. One can check (for example, this computation is carried out on p178 in Fulton and Harris's book Representation Theory: A First Course) that the kernel of this map is one dimensional and contains $e\_1 \otimes e\_1^{\ast} + e\_2 \otimes e\_2^{\ast} + e\_3 \otimes e\_3^{\ast}$.


(Note the relations from the multiplication $V(\omega\_1) \otimes V(\omega\_1) \to V(2\omega\_1) \cong Sym(V(\omega\_1))$ are precisely expressing that the variables $a,b,c$ commute, and similarly for $V(\omega\_2) \cong V(\omega\_1)^{\ast}$.)



Is the affine closure of the basic affine space of a reductive algebraic group Cohen–Macaulay?
Update: It seems that every affine closure of such a quotient $G/U$ *is* Cohen-Macaulay. In the same list of properties as above, the author states that, by a theorem of Brion, the ring of functions of $\overline{G/U}$ has rational singularities. Immediately before this statement, the author also cites [Elkik - Singularites rationnelles et deformations](https://link.springer.com/content/pdf/10.1007/BF01578068.pdf), which in particular explicitly states (Definition 1, p141) that varieties with rational singularities are in particular Cohen-Macaulay.


Unfortunately, I have not been able to track down the referenced theorem of Brion cited.


EDIT: One can also explicitly compute that, when $G := \operatorname{Sp}\_4$ and $A$ is the ring of global functions on $G/U$, we have an isomorphism


$$A \cong k[a,b,c,d,e,w,x,y,z]/(2ad - 2bc - e^2, 2bz - 2dx + ey, 2bw + ay - ex, 2dw + cy - ez, ew + cx - az).$$


where the first five variables correspond to the highest weight representation of dimension 5 and the next four variables correspond to the highest weight representation of dimension 4. This was computed according to the exercise sheet above but, as some justification for this claim (as earlier posted relations contain a typo), notice that (as can be checked in, for example, Macaulay2) this ring has dimension 6 and is Gorenstein, as expected--see [Remark 5.6 here](https://arxiv.org/pdf/1010.1606.pdf) for the Gorenstein assertion, which also gives another answer to the above question.


