>In math mode, when I select the menu item for the greek alpha 
>character, or when I type \alpha within the inline equation box and 
>then press TAB, how does lyx render this to the character?

You should be somewhat clearer what you mean with 'render'.

The general story is that each keypress will fire a LFUN_SELF_INSERT.
This will be handled by InsetMathNest::dispatch() and dispatch will call
InsetMathNest::interpretChar(). When you have typed '\alpha' and you
press space, interpretChar() will call cur.macroModeClose() which will
'finish' the macro. Cursor::macroModeClose() then creates a new Inset
with the line:

MathAtom atom = createInsetMath(name);

and this is inserted into the formula by 

plainInsert(atom);

Is this what you wanted to know ? Are you implementing something useful
?

P.S. This sounds like it is more appropriate at the developers' list.

Vincent

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