Shade-tolerant turfgrass may work in this situation. It depends on how dense the shade from the Norway maple is. Even the most
shade-tolerant of grasses likely won't be thick and lush if the shade is dense. Another consideration is the possibility of salt build up in the area. Does any of the snow come from the road (or your driveway) where ice melting salt has been used? If so, the soil structure may be breaking down. Consider leaching the salt out of the soil by watering heavily.
You also might consider planting hostas in the area. Hostas usually die down to the ground fairly quickly in the fall. (I often mow mine to the ground once they start turning yellow.) This will facilitate leaf removal and you won't have to worry about damaging them when snow is piled there in winter.