Let $G$ be a connected simply connected simple complex
algebraic group of type $ADE$ and $\mathfrak{g}$ the corresponding
simple Lie algebra. In this talk, I will explain our new algebraic proof
of the positivity of the transition matrices from the canonical basis to
the PBW bases of $U_q(\mathfrak{n}^+)$. Here, $U_q(\mathfrak{n}^+)$
denotes the positive part of the quantized enveloping algebra $U_q(\
mathfrak{g})$.

We use the relation between $U_q(\mathfrak{n}^+)$ and the specific
irreducible representations of the quantized function algebra $\mathbb{Q}
_q[G]$. This relation has recently been pointed out by Kuniba, Okado and
Yamada (SIGMA. 9 (2013)). Firstly, we study it taking into account the
right $U_q(\mathfrak{g})$-algebra structure of $\mathbb{Q}_q[G]$. Next,
we calculate the transition matrices from the canonical basis to the PBW
bases using the result obtained in the first step.
I mention also some remarks which have recently been perceived.