Questions about $\mathbb{C}[G/U^-]$ and $\mathbb{C}[B]$ Let $G = GL_n$. By algebraic Peter-Weyl theorem, we have 
$$ 
\mathbb{C}[G] = \bigoplus_{\lambda} V_{\lambda} \otimes V_{\lambda}^*,
$$
where $\lambda$'s are dominant weights. Let $U^-$ be the unipotent subgroup of $G$ consisting of all unipotent lower triangular matrices in $G$. Let $U^-$ act on $G$ by left multiplication. Then we have 
$$
\mathbb{C}[G/U^-] = \bigoplus_{\lambda} V_{\lambda}^*.
$$ 
Let $B$ be the Borel subgroup of $G$ consisting of all upper triangular matrices in $G$. Do we have
$$
\mathbb{C}[B] = \bigoplus_{\lambda \in \mathfrak{h}^*} V_{\lambda}^*?
$$
Here $\lambda \in \mathfrak{h}^*$ can be non-dominant. Thank you very much.
 A: To expand on David's comment:  let $M_\lambda$ be the lowest weight Verma for $\lambda$ an integral weight, and consider $x\in M^*_{\lambda}$, we have a function on the Borel given by $f_x(b)=\langle \ell, b^{-1} x\rangle=\langle b\ell, x\rangle$  for $\ell$ the lowest weight vector in $M_\lambda$.  This gives a map $\oplus M^*_\lambda \to \mathbb{C}[B]$.  If we factor $b=nt$, then we see $f_x(b)=\lambda(t)f_x(n)$. The function $f_x(n)$ is independent of $\lambda$ (identifying all the lowest weight Verma modules as $\mathfrak{n}$-modules), so we can think of $x$ as lying in $U(\mathfrak{n})^*\cong \mathbb C[N]$ (here I'm taking restricted dual).   Since $\mathbb C[B]\cong \mathbb C[T]\otimes \mathbb C[N]$, we see that this gives that the map is injective and surjective.  This isomorphism is precisely set up so that the functions on $G/U_-$ are those which come from $x$ in a finite dimensional subrepresentation of $M_\lambda^*$.  
EDIT: And if you're wondering what that f.d. sub is, $M_\lambda$ has by assumption a lowest weight vector of weight $\lambda$, and so $M^*_{\lambda}$ has a highest weight vector of weight $-\lambda$.  Thus, it contains $V_{-\lambda}$ if $\lambda$ is anti-dominant, and no f.d. sub otherwise.
A: This type of question is considered in the paper Longest weight vectors and excellent filtrations by Wilberd van der Kallen (Math. Z. 201, 19-31 (1989); MR). van der Kallen works over an arbitrary algebraically closed field $k$, and states his results for the Borel subgroup $B$ of a connected simply-connected semisimple algebraic group $G$ defined over $k$ (e.g., $G = SL_n$). In this context, he shows that the coordinate algebra $k[B]$ admits a filtration as a $B \times B$-module with sections of the form $P(-\lambda) \otimes Q(\lambda)$ for $\lambda$ an integral weight, where $P(-\lambda)$ is a "dual Joseph module," and $Q(\lambda)$ is "minimal relative Schubert module."
I am not well-versed on all of the terminology and conventions in van der Kallen's paper, so will leave it to the reader to consult van der Kallen's paper for the precise definitions of the relevant modules appearing in the filtration. I'm also not sure how this description fits with Ben's description when $k = \mathbb{C}$.
